I'm around 10,000 kilometers from my copy of the Climax of French Imperialism: 1914-1924, which makes it hard to check for precise details in some sections, but the French were profoundly unhappy about the break-up of the Ottoman Empire. To the French, from their perspective, losing their influence across the Ottoman Empire as a whole - and getting in return Syria - wasn't desirable. The Comité de l'Asie Française for example, saw de Caix, their main ideologist, saw the following: "If means still exist to save the Ottoman Empire, we must seize them . . . we prefer to make our contributions to the cultivation of the great Ottoman garden rather than to have our small plot in Syria to ourselves". Delcassé and Bonpard (the French ambassador to Constantinople before the war), were both opposed to a partition. Most French high diplomats seemed to have shared a general view, this being the intent of the Quai d'Orsay. To take some quotes, unfortunately the Climax of French Imperialism doesn't generally mention exactly who said them...
"Our moral and beneficent influence would be severely limited, perhaps ruined, by a partition of the Empire."
"Everything torn away from Turkey is also lost to the French language . . . We can scarcely hope to find in the Orient, outside of the Turkish empire, Turkish or Arab authors who choose our language in which to write and who sometimes use it with such genuine talent . . . If Turkey were to be disassembled then the loss to our cultural domain would quickly become irreparable."
"[the official policy of the Comité de l'Asia Française was to preserve the Ottomans intact as]One of the most favored areas for our economy activity and-more important still-for our culture."
Only the diplomats in Syria and those who were concerned that the British intended to steal the region from them (quite a correct concern) dissented - and the latter were mostly invigorated when the war began. The French seizure of Syria and Lebanon is the French making sure they got their pound of flesh as their second-best option, rather than necessarily being the over-riding goal for them in the region.
I doubt therefor that the French would be likely to support a partition of the Ottoman Empire, and certainly wouldn't proceed therefor on their own. If the French can, they'll prefer for the Ottomans to stick around. Russian relations with the Ottomans had been quite good a few decades before, and it isn't impossible that they might return to this. French diplomatic pressure on the Russians would help in this regard; the French will still hold those massive loans in Russia and serve as a source of Russian financing, giving them outsized political influence there (and on a similar note, their massive loans and influence inside the Ottoman Empire, of which some might be relinquished by the Ottomans joining just like Siam saw some reprieves from Franco-British imperialism but which much would remain intact, give a very good concrete reason to support keeping the Ottomans together). I don't know what the British relations to the Ottomans were like, but I doubt that the British are going to invade the Ottomans. Greece, Serbia, and Romania attacking the Ottomans seems bizarre... what do the Serbians post First Balkan War, much less the Romanians, get out of it? The Serbs happen to have both Bulgaria and Greece between them and the Ottomans, and no Serbian minorities in Ottoman territory. Greece on its own, has little hope of invading the Ottomans.
Italy meanwhile, while fully capable of biting off disconnected Ottoman territories in Libya and a few islands, seems like a... doubtful, candidate to mount an invasion of the only Ottoman territory of significance remaining, the mainland.
This being said, the French did have that second option of partition, and it existed for a reason. I think the best quote to sum up what French policy is, is one from Raymond Poincaré:
"We must maintain the status quo in Asia Minor as long as possible. But there will come a day when partition takes place . . . and we must make advance preparation in order not to miss out on it."
If the Ottomans look strong, then France will back them; if not, then the buzzards will circle.
