A noon or even 1:00 pm flight from STT gives you pretty much no time to do anything on the island that morning unless you're staying right by the airport.
I would ask for the evening flight to MIA, overnight in MIA, and take a morning flight to STL. There are plenty of very nice but inexpensive airport hotels in MIA, plus it'll still feel somewhat tropical - unlike ending a Caribbean vacation with a night in NYC in December.