Lincoln would not do this but I can believe other Republican leaders, or perhaps an ATL reshuffling of the party that holds the Republican party's place, might do it. OTL there were Republican aligned factions suggesting "Go in peace wayward sisters."
My sense of how it works out is that the Confederacy (which strictly speaking need not form as a rival Federal government but I think almost certainly would, perhaps in somewhat different form--I suspect on paper, in terms of its written Constitution, which was nearly identical to the US Constitution OTL, it would be the same, and differ in actually attempting to operate by it rather than be overridden by the war emergency) gets very little beyond the maximal count of secessionist states, and even a more wishy-washy President than Lincoln would make efforts to hold some of the border states. It would be damned awkward to let Missouri slip into the Confederacy for instance; going around it would be hard. As OTL Kentucky might well stay in the Union, and Maryland I think would be held forcibly, as would Delaware (where I am told the slave interest was weak in any case). Thus the Confederacy would border the USA on the Virginia border, all the way round to Tennessee which I guess would go Confederate, thence to Arkansas.
A major question is what happens around the Indian Territory, by this time confined to later Oklahoma. A common trope going back generations is that the Natives there ally with the Confederates. I think that is a bit under-nuanced; some tribes were slave owning and historically OTL did connive with the rebels,but others would not be so inclined--it could well be that the pro-Confederates, with Southern help, overwhelm their opposition and in effect OTL OK is also Confederate. But I don't think the South would manage to seize much territory farther west. Certainly they tried OTL, but they actually did not have all that much impressive success, considering that the Union could spare little manpower or other resources to try to check them--locally raised militias in New Mexico were able to accomplish this remarkably well.
Assuming some kind of Alt-Republican in Washington does adopt a "let them go" policy, it won't apply to any Western territories, and this President would surely be able to recruit enough Unionist-loyal troops to reinforce Santa Fe and make damn sure the Confederates don't seize any of that Territory or California. Not inconceivably, a very clever Confederate policy might make allies of the Mormons and also selected Native peoples--but the trouble with that, aside from the question of how likely the Mormons at least would be to make such an alliance, is that any messing around of this nature is an act of war. Having been conciliatory enough to let the secessionist states go (mostly) there is no political capital in allowing the Confederacy to pirate away anything else. Whether the Confederacy might turn to new conquests in Mexico south of the USA border, or say invade Cuba or Nicaragua or some such, is another matter (FWIW I think they'd tend to overestimate their abilities quite badly--perhaps even so, have enough overwhelming force to prevail, but it would cost them a lot more than they'd arrogantly reckon, initially and trying to hold their conquests long term).
So Oklahoma might be a loss, due mainly to large numbers of Native Americans having ample historical reason to mistrust Washington--but how can they better trust the Confederates? (There is, again, some OTL evidence some would anyway). From a humanistic point of view, I'd hope that this hot spot on the border would lead to the Union adopting a conciliatory and principled policy toward at least some Native groups who demonstrate pro-Union loyalty, and reward them for it. Along those lines, early statehood for New Mexico, to shore it up as a firm doorstop on Confederate designs on any further US territory, along with reinforcement of its military defenses and occupation of Colorado would also be in the cards.
Without the Union making war, it could be as noted there might not be a Confederacy at all, though I certainly think Virginia would seek it as otherwise it is their single state facing most of the most dangerous borders with the Union. The capital might not move to Richmond if there is no war. (I suspect Montgomery would prove somewhat unsatisfactory, due to poor communications, but given the philosophy of strong states and a weak federal governments the secessionists professed, they might like a weak capital just fine.
Texas might not join the Confederacy, and there is some chance it would not even secede--Sam Houston was dead against it for one. But given that the aforementioned attempts to drive a corridor to the Pacific were essentially Texan actions OTL, I fear that secession at any rate would be successful (and if Arkansas secedes as OTL, and Oklahoma can be captured via the help of strong factions among the Native peoples there, then Texas is largely insulated from any direct contact with the Union. Texas could therefore well afford to be independent, assuming no commitment in Washington to retaking all secessionist territory. Vice versa, the factions most in favor of secession probably do cherish expansionist notions of some kind--initially they'd try as OTL to take some Union territory but I think they'd be rebuffed, and the rest of the secessionist states, Virginia especially, would be twisting Texas's arm as hard as they can diplomatically to get them to back off. Whereas if the expansionists resolve not to prod the Union sleeping giant and stay south of US claims, I'd think Texas would want the backing of the other secessionist states in money and manpower for a shared Confederate strike into Mexico or overseas. Might the CSA negotiate a partition deal with Maximillian, aiding his bid for a throne in most of Mexico in exchange for alienating the northernmost tier of states and territories to give the CSA a Pacific port in Sonora? I believe if the CSA is granted this tract, they'd be occupying the zone Juarez fought Maximillian from, or one of them anyway.
The basic reason I think "no war" means the Union keeps all territory not belonging to a secessionist state, barring a portion of Indian Territory perhaps, is that for the CSA to claim any of it would require acts of war. The premise that a Northern Republican government in the USA wants to avoid the terrible cost of the Civil War does not commit that northern government to total pacifism or helplessness. The South presumably also has reservations about the desirability of getting into a knock down fight with the North, and persistent violence on the frontiers risks that.
Vice versa, the USA has no leverage to for instance carve off West Virginia from the now-Confederate state of Virginia. To assist the mountaineers seeking to escape Tidewater dominance in western Virginia's uplands would be an act of war just as much as Texas trying to rip of part or subdue all of New Mexico. It might be possible for the Union to conduct good faith, above board negotiations with the Confederacy and/or Virginia for a peaceful consensual partition--probably not including the full sweep of West Virginia of OTL; the piece that Virginia might be persuaded to give up would probably be mainly the long northward salient, which would have to be annexed to one or the other neighboring state I'd think.
Along those lines, it might prove difficult and politically unpopular to try to hold on to Maryland. Aside from just giving up and letting Maryland join the Confederacy, another option might be to negotiate Maryland becoming a nominally independent republic as a neutralized buffer state, with treaty rights to negotiate trade terms with both the USA and the CSA. Might Kentucky go the same way? Perhaps along with a strip of southern Missouri.
Obviously in that case, and anyway I think, it would be foolish to try to maintain the Union federal capital in Washington DC. For a time it would be necessary to stay there, due to the heavy infrastructural investment in the capital, but this was not nearly as deep as it would become in the 20th century. I would think a new Union capital district should be carved out in the Midwest (the eastern part of it actually, Ohio might be too far east but Chicago would be too peripheral at this date--to put a finger on where I'd want maps and a lot of detailed information about trade volumes over river, canal and railroad as of this date, as well as the input of someone regarding strategic defensibility. Moving it back from the border though seems an obvious necessity.
In these kinds of threads you always get someone asserting that the Union would find loss of control of the lower Mississippi "intolerable." But if so, then the Union must go to war. Tennessee, the state of Mississippi, Arkansas and Louisiana are all secessionist, and only denying them the "right" to secede can result in the Union controlling that river. Given a decision by the Union government to avoid the Civil War, it is a done deal; the CSA owns the lower Mississippi river and the Gulf ports. But it would hardly be in the interest of persons making money off river traffic to close off trade with the North! Had the secession crisis come a generation earlier this would be a tough problem for the Union, and certain pundits of the day OTL asserted that allowing the South to hold it would force the West to join the Confederacy, but between the vested interest southern shippers would have in enticing Yankee trade down and up the river, and the fact that the northern states now can turn to railroads and northern watercourses, sending traffic to the Great Lakes and St Lawrence (or via canal and RR, to the old Atlantic northeast) I don't much credit this threat.
What is more of a threat, long term, is the simple fact of a long land border with another government, along with the fact that the southern confederation is committed to slavery. People argue back and forth about this--I believe southern slavery will die unreasonably hard, because while in the abstract one can argue that "slavery is irrational," that maxim first of all is not absolutely and categorically true--slavery was very very profitable and foundational, not just of the USA but the whole Euro-American system.In any case, the secession was hardly an expression of the uniform will of every southerner--it was rather the pet project of a certain faction, and that faction was deeply intertwined with the slave interest. The current ruling class in all the secessionist states is slavocratic. It may be that over time this order will erode or be forcibly overturned--but that is hardly a prophecy of stable peaceful relations with the CSA!
Will Northern society evolve to keep the peace or to clash violently with the South? I certainly think elements within it will maintain a cry for vigilance and a certain degree of hostility--it may be these elements will not ever prevail, or moderate before they can prevail. One hard dynamic at work is that as long as the South maintains slavery, slaves will seek to escape, and perforce try to cross the border into the North. It is possible that by treaty, the Union will agree to continue hunting down and repatriating fugitive slaves--but actually one of the huge tensions leading to the Republican party forming and dominating northern and western elections was outrage at that very practice.
Even if the ATL Republican president is as conservative or more so than Lincoln on the question of African-American citizenship and whether slavery can be tolerated on any scale, if the US signs a treaty with the CSA agreeing in principle to repatriate slaves, you can bet there will be damn little inclination to actually enforce the US side of it. It would be largely if not entirely a dead letter. The teeth of Fugitive Slave laws prior to secession related to the right of Southern interests to send devoted agents physically into the North to appear at courthouses with warrants for capturing specific individuals which Federal law and court rulings said state and local officials had to honor.
So actually I would expect the Union position to be that secession is a divorce and the Southern whites have zero claim on Union policy, and have no rights to have anyone remanded back to their custody. The major avenue of recourse Southern interests might have is playing on Northern racism, and getting majorities in the North to favor a white-only policy and attempt to ethnically cleanse the North of its already present African American people.
But in the context of Southern secession, I expect that there will be voices raised among "white" Northerners on behalf of the African-American persons among them.
In the context of the CSA merely existing as a potential military threat, there will be a desire to beef up the Federal military, Navy and Army, and African-American refugees stand ready to join these expanded services. Their personal interest in the Union cause would be quite strong and reliable after all. I believe at least some political factions will favor citizenship for AA persons, and value their usefulness, and over time they will integrate deeply into Union society--as many had long done already.
Thus there would be a faction--AA persons and their political allies--who will have serious issues with the CSA. Even if slavery as such collapses fairly soon (as it hardly can do without major political earthquakes, unless it is a managed transition to another form of deep oppression--setting the former slaves up as a subclass of peons with "no rights a white man is bound to respect" still perhaps, or collective state slavery, or private chattel slavery made more useful via massive violations of personal rights and dignity beyond the already terrible facts of US law before the ACW (using drugs or brain surgery, or sophisticated forms of torture along lines of operant conditioning, and so forth) we can be sure the condition of AA persons in the South will be pretty nasty (unless perhaps there is a deeply radical revolution with "whites" as close revolutionary allies). And persons who originated in the South will have unsatisfied interests in safe return being barred to them, and connections with kin and other folk they had known in the South.
In the immediate secession crisis, the rational US policy would be to mend fences with Britain. OTL Republicans often muddied these waters with wild talk of Canadian conquest and so forth, but an administration that wants to avoid violent civil war with the South is hardly going to seek a war with the world's greatest power instead--particularly as the British could assist the Confederates, or anyway the northern war would be a distraction making CSA initiatives on the southwest borders more feasible. The smart thing, having bid for peace, is to keep the peace--with Britain especially. Having shaken off most of the US slave population, the USA is no longer at odds with Britain over abolition. With lower Mississippi access to the Gulf of Mexico under CSA control the USA needs to be sure of good access from the Great Lakes to the sea via the St Lawrence. Coordination of development policy on the northern border is now a major priority.
Given the way US and British interests evolved OTL, I don't think there is any reason to doubt the USA will grow at least as close and perhaps far closer. There is no reason for Britain to aid the Confederacy with any great preference, and the Southrons could easily become obnoxious in the Gulf and Caribbean and Central and South America. Or of course they might instead pursue superior diplomacy than the often brash or rude Yankees, some of whom do persist with the talk about taking Canada by force and a lot of general Anglophobic rhetoric--some traditional 4th of July stuff, some from factions like Irish immigrants on very general principles. I'm just saying here the usual trope about Britain joining with the Confederates at the hip and the USA therefore allying with the Germans or some such is hardly a given nor particularly likely--on a rational basis, the northern Union has more to offer in partnership than the southern one.
Certainly there is no "Trent War" pretext if the USA never fires a shot at the Rebels in the first place.
