Flames of Paris seems to me as peculiar a choice for ABT (even in the Ratmansky version) as Bright Stream. Those ballets have a reason for being in Russian companies; much less so in an American one. The full-length Ratmansky that I thought might be imported by ABT was Lost Illusions.
I don't think ABT would do Flames of Paris well. It would probably repeat the fate of The Bright Stream, which was declared a hit in its first season and played to half-empty houses the second time around. Lost Illusions would seem to be a better fit, but it is, I think, hampered by its amorphous music and dramaturgically: the hero and heroine don't intersect in the last act.
I think, Ratmansky's Paquita ,which is being premiered in Munich next month, would be a better fit for ABT than Flames of Paris. American dancers can't do well all those character dances.