Assuming that the fall is prevented by the lack of an alliance with Mongols : badly.
Latin States in the latter period became more and more italian protectorates (or at least their
battleground), and the process was quite advanced under Lucia de Sergi's regency, or present in others part of Antioch-Tripoli (as
Embarici lordship).
Eventually, I think Antioch-Tripoli would be in a similar situation than Jerusalem, with a barely reigning prince and the country ruled by nobiliar councils more or less influenced by mercantile italian interests.
Admittedly, Antioch-Tripoli would have a more important Armenian influence, but I don't think it would counterbalance the situation but rather add to the general mess as the Armenians in the
county of Tripoli (that is the continuation of Antioch-Tripoli at this point).
Of course such situation would likely prevent it to take Damas, wouldn't be for Baibars power.