There were Caribbean colonization attempts OTL. They tended to fare somewhat poorly. But even if you succeed, you get the same problem as Liberia: the "colonists" have no real connection to the people they're settling among and they don't necessarily have the skills to adapt readily to the new conditions. There's a reason Liberia is not exactly a shining paradise, even by modern African standards.
I considered this, but there would be very real differences between Liberia and Haiti in this case. Liberia was a true colonization. Freetown and Monrovia simply did not exist before the African American colonists arrived. They were not joining into existing societies, so the Americo-Liberians immediately became a distinct ruling class isolated from indigenous populations. The difference would be comparable to white settlement of Native American lands vs. the American immigrant experience.
Also, Haiti falling within the US sphere of influence would offer some protection from European imperial ambitions that Liberia did not have (admittedly while obviously opening them up to American imperial ambitions). Proximity to the US would also mean Haiti would be significantly less isolated than Liberia allowing easier assistance from America. In particular, I could see black technocrats educated in American universities moving to Haiti to help the country make necessary agricultural, land management, and political reforms that would be crucial to increasing its chance of success. Would they relocate? Would they change Haiti rather than be changed by Haiti? That is much more speculative.
Again, this is a long shot, but I'm not convinced that keeping Toussaint fundamentally changes things. I think he would have ruled much in the way his OTL successors did resulting in the same economic mismanagement and political instability. In reality, the social situation prior to independence was just so poisonous that the country's future seems bleak with practically any roll of the dice.