I don't think that you will find a true AI (unlimited alcohol and diving included) in the Caymans except the Aggressor liveaboard that includes beer and wine, but there are some nice places that include food, lodgings, and diving, and offer some drink package options
We bring a bottle of our favorite liquor with us and mix many of our own drinks and may supplement with a few drinks from the bar. Alcohol is expensive in Cayman. Before we leave the US we stop at the duty-free store at the airport and each adult buys a 1 liter bottle duty free to carry on the plane. If GC is your final destination that isn't a problem, but if you are traveling on to Little Cayman or Cayman Brac then you will have to go through customs and immigration on GC, and then recheck your bags and go through security before boarding a flight for the sister islands - and security will not let you carry any liquids greater than 3 oz. and they will take your bottle away from you.
So you have to put your bottles in your checked luggage before rechecking - it's only a short flight and you can find good, reusable liquor transport bags on Amazon to keep them safe. At least that is how things work right now, but the GCM airport renovations are supposed to be finished by the end of this year, so things may be easier by next June.
At the Clearly Cayman resorts (Little Cayman Beach Resort, Cayman Brac Beach Resort, and Cobalt Coast Grand Cayman Resort) they offer alcohol packages for 3 drinks per night, but I have never used them and they seem a bit cumbersome, but others can tell you more. You carry a little card to check off your drinks, and I think that your selection is limited, but I am not sure. Ordering drinks al a carte is pricey.
We've stayed at the LCBR and the CBBR and we have visited/dived with the CCGCR, and we have a good opinion of all of these properties. They are (almost) all-inclusive because the price includes accommodations, airport transfers, meals and diving, and the hotel taxes, and service charges are also included, but not the dive shop tips. Nitrox is usually extra everywhere in Cayman.
Regarding LCBR and CBBR, the resorts and accommodations are lovely, the diving is beautiful, and the food is very good, and you can have fun around the beach bar every night with the games and entertainments offered. There are outdoor ice machines near the bar that guests can use for their own beverages and small refrigerators in the rooms. You can buy mixers like soda at the dive shop or at local stores (but not on Sunday on Brac). Don't buy mixers from the bar because they will add on a service charge. There are bicycles available for the use of guests.
I think that the diving is best at LCBR and the food is very good; while the food is best at CBBR and the diving is very good.
We enjoy both islands and resorts. There is more natural beauty to enjoy on Brac but they are both lovely, quiet, friendly, little islands. Both resorts have nice beaches for sunning and relaxing but they are not good for swimming or snorkeling because there's turtle grass, rocks, and dive boat traffic. Both resorts have pools and hot tubs and the infinity pool at the CBBR is huge and gorgeous. Both resorts have small spas onsite that are very nice and offer a variety of services. You won't need to rent a car on LC or CB, but we usually get a car for one day just to explore. Traffic has never been an issue for us on LC or CB.
There is also the upscale Southern Cross Club on Little Cayman and it is a lovely property with a meal plan. They have an onsite dive op and fewer guests, but it is more expensive than LCBR. We have never been to Pirates Point on LC, I understand that they have excellent food but a limited, daily pre-fixe menu, and you have to be transported by van to the dive boats. I think that they only offer 2 dives per day at PP.
The CBBR is the only dive resort on CB, but there are some condos and rental properties, and there is a second, well-regarded dive op, the Brac Scuba Shack. There is also shore diving available on CB but we haven't tried it. There are some restaurants and grocery stores on Brac, but not a lot of choices, and food is expensive.
Our knowledge of CCGCR on Grand Cayman is more limited because we have never stayed there, just visited and dived with them. The property looks very nice and they are currently completing a lot of renovations that should be done this year. My understanding is that they do not have an indoor dining room, so you eat outside or at the roofed bar, and I am not crazy about that setup; but it's probably fine.
Some of the advantages of CCGCR are that they have a house reef with unlimited shore diving and snorkeling onsite, and you are close to the North Wall and that is very good diving. The shore diving can get winded out, but June is not typically a windy month in Cayman.
Another advantage of CCGCR is that you are on Grand Cayman and there are many attractions and venues to enjoy - but you will need to rent a car because CCGCR is hell and gone from everything else on the island - and I mean that literally because it is located close to the town named Hell (aka cheezy, tourist trap). The CCGCR is situated on ironshore, but they have a pool and a small man-made beach. I don't think that they have a spa but there are lots of spas on GC.
The Reef Divers dive ops are excellent at all 3 Clearly Cayman resorts and they offer nice boats, good staff and facilities, and valet services in that they haul your gear and help you into and out of it and they take care of your BCD, regulator, and weights. You just have to rinse and hang your wetsuit, and mask, fins, and snorkel - and of course take care of your camera and non-integrated computers. I am not quite sure how the gear handling works at CCGCR because of the shore diving option.
Other (semi) all-inclusives available in Cayman include Compass Point on GC - because I think they offer a meal plan and they have an onsite dive op - and maybe also The Wyndham Reef. They are both on the East End which has great diving but they are very far from everything else so you will need a car - just getting there from the airport would be a very expensive ride for your party.
Of the two, Compass Point is the more dedicated dive resort - but no beach, but there is a beach at The Reef. We've eaten at The Reef and the food was okay but nothing special, and there are so many good places to eat on GC, and it is safe to explore the island, so I probably wouldn't take a meal plan. There are far fewer eateries or grocery stores on LC and CB so a meal plan makes more sense in the sister islands IMO.
The Holiday Inn on GC has Divetech as the onsite dive op and I think they offer a meal plan and they also offer shuttles to various attractions - at least during the day. You also get unlimited shore diving at Divetech's Lighthouse Point Facility if you stay at HI, but you will need a car. I don't think there is a beach at the HI, but I am not sure.
Sunset House on GC has already been mentioned and breakfast is included. They have an onsite dive op for boat dives and unlimited diving on the house reef. There are 2 restaurants onsite and it is close to George Town. If there are photographers in your group, they will enjoy the Cathy Church photoshop, studio, and training center onsite. You could avoid renting a car at SH because transfers are included and you are less than a mile from George Town and the buses run out front. There is a pool but no beach at SH.
Sunshine Suites is in the 7MB area, but across the street from the beach, but it includes full access to the amenities of the Westin Hotel and beach. The Sunshine Grill is the onsite restaurant and you get full kitchen facilities in the suites. There is a dive op at the Westin, but you can dive with many ops if you stay in the 7MB area. They will pick you up by van or by boat on the beach. Everyone has their favorite, but in general the dive ops on GC are high-quality, there is too much competition for a bad operation to last very long. There is also good shore diving available on GC.
With your large group, if you want to stay on GC, you should also look at Coconut Bay Condos. They are on the West side of the island, not far from Seven Mile Beach, but north of 7MB. They offer 2 and 3 bedroom condos and have an onsite dive op and shore diving on the house reef, but no beach. You will probably need to rent a car but Cayman does have a good bus service, at least around 7MB and GT, primarily during the day. You can prepare some meals and you will be near to many places to eat; especially if you rent a car.
If your party is primarily interested in diving, then you may want to go to LCBR or CBBR; if they want more topside activities and attractions then consider your options on GC.
If you do decide to go to one of the sister islands, try to add on a couple of days on GC and stay someplace on 7MB, like the Comfort Suites, to check out the island and see if you'd like to return.
Reef Fanta-Seas are the booking agents for the Clearly Cayman resorts, and they are located in St. Petersburg, FL - which I think is near your location, see below. You might try dropping by and talking to them in person; I am sure they would be interested in booking a large party like your group during the slower month of June - and they may give you an attractive offer. The big Long Beach, CA Scuba Show is coming up very soon, and they will probably be offering a "show special" so ask them about it and they may offer it to you, too. Good luck!
Reef Fanta-Seas
Reservations Office
4127 5th Avenue North
St. Petersburg, Florida 33713