Originally Posted by flyerwife
Grand Cayman Marriott.
Others will disagree with me (do a search, lots hate it) but I liked it a lot and it fits all your criteria perfectly.
I stayed at the Grand Cayman Courtyard and liked it.
My favorite island for a nice, quiet family vacation is Grand Bahama Island (Freeport, Bahamas). Unfortunately, there are no Marriotts there.
For the OP, also take a look at airfares. We flew on Spirit to Grand Cayman and it wasn't too expensive. The airfare to some Caribbean islands can be terrible.