Air Jamaica flies Santo Domingo to Montego Bay and Kingston on various days via Port Au Prince Haiti ( I think the same plane though so it should be reasonably safe) .
I am not surprised that there is nothing from POP itself - I think you are stuck doing POP-MIA then MIA to either Jamaican destination.
You can do Caribe/Metro bus POP to SDQ in a reasonable ride and at least via PAP it is sort of all going in the same general direction.