An interesting idea, and one that does have a few potential routes that I haven't seen explored before. The obvious one would be the 6 February 1934 crisis turning into a March on Rome style situation, leading to the Far Right coming to power and restoring the monarchy, but that's kinda boring (and would require the monarchy surviving being restored by the Facists). Instead, I'd focus on something in the 1940s-1960s, around Charles de Gaulle and the Comte de Paris. I'm gonna quote this directly from the Comte's page on Wikipedia:
"
In 1954, Henri met Charles de Gaulle and continued their relationship through correspondence.
[14] In 1958, Henri gave his support to de Gaulle, who was called back from his self-imposed exile to save the French Republic from insurrection in Paris. Thereafter, Henri became a frequent visitor to the
Élysée Palace, where de Gaulle waited for Henri "by the staircase or outside, reserved a special armchair for him and lit his cigarette." There, they frequently discussed French history together, with Henri noting that de Gaulle loved to pronounce the word 'king'.
In 1960, de Gaulle told Henri that "Monseigneur, I believe deeply in the value of the monarchy, and I am certain as well that
this regime is the one best suited to our poor country."
[12] The following year, de Gaulle dispatched Henri on a tour to
Libya,
Ethiopia,
Iran, and
Lebanon, with the purpose of explaining France's Algeria policy, serving as de Gaulle's special representative, or "pro-consul." During this time, Henri befriended
Hassan II of Morocco and
Habib Bourguiba. In 1962, de Gaulle informed Henri in strict confidence that he had arranged the French presidential election so that the head of the royal house could succeed him as president of the Republic.
Georges Pompidou confirmed this, telling a close friend that "I know the general has made up his mind in favor of the count of Paris." However, by 1964, de Gaulle changed his mind and told Henri of his decision to run for re-election, which he won. By 1968, Henri ceased publication of his paper over his increasing disagreements with the Gaullists.
The Countess of Paris remarked that "Under de Gaulle, Henri came two fingers close to becoming king. But by 1968, it was all over, finished."
So going by the above, there's definitely an opening for the Comte to emerage as de Gaulle's protegé and political heir, being elected as Prince-President in 1965. If this happened (the heir to the throne democratically elected by the French public as Head of State), I can easily see this lead to a restoration of the Monarchy. Now, I have no idea how this would effect French politics or if this would mean a Kingdom of France in which the Crown has similar powers to the modern President, but its a very interesting scenario to go down. And, before anyone says its impossible for Paris to win an election, remember that he would be backed by the ultra-popular President de Gaulle (who easily won reelection in 1965) and the Gaullists (who remained the largest party in Parliament from the 1962 election). Moreover, there is an existing, modern example of a Prince winning a democratic election; between 2001 and 2005 Simeon II, the last Tsar of Bulgaria, served as Prime Minster of the country after his party won a massive majority (defeating both dominant political parties).
Another idea is the Comte de Paris accepting de Gaulle's invitation to London in 1940/41. Again quoting from Wikipedia:
"Between 1940 and 1941, the Gaullist camp offered Henri an invitation to go to London, which he declined. Henri feared that if he accepted the offer, he would have become an
émigré, like the Bourbons who returned to France after
Napoleon's defeat. Henri was staunchly opposed to the idea of siding with one political party, wishing instead to pursue a path of unity and not contribute to France's "infernal divisiveness."
[9] Charles de Gaulle later confided to his biographer, Phillipe Saint-Robert, that "Had the count of Paris joined me in London in 1940, he would have become France. Together, we could have done great things."
So, going by that quote, a Comte de Paris that got over his fear of being an émigré could easily emerage as a hero of WWII alongside de Gaulle. What that means for the post-war government, I do not know. Could mean a third restoration instead of a 4th Republic, or a boost for him in a later '50s and '60s TL.