We travel in a group of 7-10 often so were interested in charters. Until we did the math.
At DNS they have package rates of 10 dives/$500. Or charter rates at $850 since they'll take 8.
I like Wall to Wall Diving - know the owner casually.
They do 10/$540 or $650/charter. So more reasonable but still more expensive.
Personally what I'd do is book daily diving with one of the 6-8 divers max operators. Far enough out so they don't have any other divers booked. On some it would be like a private charter anyway - minus lunch - although they all provide fruit/water etc. FYI Indigo Divers larger boat maxes out at 6 divers. It might even be a little crowded.
Other options I'm pretty sure limit diver load are Ambassador, Neptune's Divers, Off the Wall and Deep Blue. The latter two are based in Georegetown so that's no fun with the cruise traffic daily. Although they work out of Lobster Pot's dock which is almost far enough north to be less affected.
Often on Cayman boats they take suggestions/vote if conditions are decent. So your family would "own" that most days. Watch some of the videos on
Dive in Cayman, Grand Cayman Diving, Cayman Islands Scuba Diving to get a good idea of sites.
Here's a pretty complete list of dive operators in your search for others.
Cayman Islands Tourism Association - Things To Do In Cayman, Cayman Culture,Cayman Islands Watersports
Some that are probably too large are Fosters (cruise contracts and downtown), Red Sail (cruise and multiple locations) Sunset House (resort guests and only morning dives except for afternoon specialty dives) and probably Divetech unless they added to their fleet since they now support 4 resorts with the same two boats they had at Cobalt Coast afaik.
Lighthouse Point is a good shore dive, so is Turtle Reef nearby. With as many divers as you're paying for it's a consideration..lol. Sunset House is also worth an afternoon, there's two decent dives - the Mermaid and the LCM wreck, the Cathy Church photo gallery/store and MyBar is waterfront for inexpensive food.
I'd bring the gear also. Complete rental package rates range from $80-125+/wk. I don't know of any shop that includes it. Many of the operators are dock based so storing it is an issue for them as well.