I would explore some of the options in Cane Garden, not far from the west end where by that time hopefully, if not already, some charter boats depart from. I have stayed at Lighthouse, Mongoose Villas, Columbus Sunset and Quito's before the renovation. Columbus has 2 bedroom options which might be good enough for your family. Down the street from the bar scene on the beach, so you can get away, but close enough to walk every night to a few different options. I always try to stay there for a few nights when I am in the Islands.
I have not stayed but would love to, at Agape Villas, also a great option with some larger rentals for a family like yours.