No kids here, but some places to consider for dining that may work:
George Town Yacht Club - has a small pool and especially popular with families on the weekend.
Sunshine Grill at Sunshine Suites
Periwinkle at the Ritz - for family dinner & movie night
EATs - similar to an East Coast US style dinner.
Ortanique at Camana Bay
You can find info and menus on most places on Grand Cayman here:http://www.caymangoodtaste.com/