Orleanist. Most monarchists in the 20th century were Orleanists.Well mainland french monarchism would be interesting to see but what will the people accept? Bourbon? Orleans? Bonaparte?
The Bonapartists pretty much became unpopular after losing Alsace-Lorraine. Given that its loss was a domino effect to what led to their current predicament ITTL, coupled with the Bonapartes being pro-Free French IOTL, which I can see being the same case here, support for them is probably near zero.
As for the Bourbons, they aren't palatable to monarchists outside of a few who are in die-hard opposition to the Orleanists. The reason being that the head of the House of Bourbon is a Spanish prince (at the time the second son of former King Alfonso XIII)