I'm not sure when he tried to make Utah a slave Territory.
The following is from an interview given in 1859 with Horace Greeley.
http://wiki.nycldshistory.com/w/1859-08-20-New_York_Tribune-Interview_with_Brigham_Young
...
HG: Are there any slaves now held in this territory?
BY: There are.
HG: Do your territorial laws uphold slavery?
BY: Those laws are printed -- you can read for yourself. If slaves are brought here by those who owned them in the States, we do not favor their escape from their owners?
HG: Am I to infer that Utah, if admitted as a member of the Federal Union, will be a slave state?
BY: No, she will be a free state. Slavery here would prove useless and unprofitable. I regard it generally as a curse to the master. I myself hire many laborers and pay them fair wages. I could not afford to own them. I can do better than subject myself to an obligation to feed and clothe their families, to provide and care for them in sickness and health. Utah is not adapted to slave labor.
...
There isn't any doubt in my mind that Brigham Young was the most anti-negro of the Presidents of the Church (in absolute terms, if relative to the US population at large that point, there might be an argument for Harold B. Lee), *but* if statehood was offered to Utah (with pretty much any boundaries that includes SLC, as large as the proposed State of Deseret or as small as what ultimately became Utah) with the requirement that it be a free state, Brigham Young would have taken that deal in a heartbeat.
But by 1856, the linking of Slavery and Polygamy as "The twin relics of Barbarism" in the Republican Party Platform and the passing of the Morrill Act in 1862 put the Federal Government (dominated by the Republicans at that point) on the path that it took iOTL.
I'm expecting that the events of a more radical civil war may lead to actions against Polygamy and the Church being harsher faster. If the Morril Act is ineffective as OTL because Utah juries won't convict, the replacement laws and court decisions from the late 1870s and 1880s may occur earlier (while BY is still prophet, he died in 1877)
I haven't seen many TL where the 1860s to end of Polygamy go more smoothly than OTL, but a lot where they go worse.
In some ways "The twin relics of Barbarism" are like George W. Bush's "Axis of Evil". Just because you don't align with multiple things doesn't mean they get along with each other.