I think the land would become American somehow – the French were in no place to exploit it (France used its surplus manpower in European land wars, and its major settler colonies were in North Africa), especially compared to the Americans, especially in the face of the Americans.

Which is not to say they were helpless and we coulda yoinked it for free. They coulda stirred up and equipped native proxies against us, they coulda blockaded New Orleans until we paid (like how they got Haiti to pay for its emancipation) they coulda landed a punitive expedition and burned our capitol like the British.