Nearby - Ambassador Divers, Neptune's Divers or Indigo Divers. All dive smaller groups only.
But they're all good -
Cayman Islands Tourism Association - CITA Member by Watersports/Dive Sector
Red Sail is at the Westin - may
not be what you're looking for - depends on the day. They're the Disney Cruise op.
Foster's has the rest of the cruise contracts plus they're south of town so maybe not them either.
Everyone is 8 divers/DM - some that's groups, others that's all they take per boat. Many specify on their websites. 4 of you might be considered a full load with some of the smaller ops. I've heard, but not experienced - that Ambassador could be one of them.
Just about every shop will pickup free at most resorts/condo complexes near the Westin.
Dive
Turtle Reef also - it's an easy entry shore dive. Lunch at Macabuca Tiki bar after. Short drive north from the area you're staying.