By dive shop at your hotel, you mean you want to be able to have shore diving from your hotel, or the boat leaving from your hotel? Sounds like you probably realize that "beach" is often incompatible with diving from the hotel, since "beach" often equals "shallow bare sand a long way out." Cayman has some great diving but not so many options for diving on site as someplace like Bonaire. Lots of people's dive days simply start with hopping on a shuttle or driving someplace. (This is often desirable, say you want to go to the North Wall, they don't take boats all the way around from the other side, they will bus you the short relatively short distance across the island to a dock on the other side where they keep a boat, for shorter boat trips.)
I think there are a few ops on Seven Mile Beach where they pretty much pull the boat up on the beach to pick you up, but wouldn't be my choice - this is the more congested area of Cayman and they would be doing west wall diving from there which tends to be crowded and is not as good anymore as the other areas. And you don't shore dive from SMB hotels, too far to swim to anything. You can shore and boat dive from Sunset House, but its ironshore, no beach. You can usually shore dive and occasionally boat dive from Cobalt Coast, but again no beach. (Hanging out by the pool on the water is nice though.) IMO CC is a much nicer place and better dive op than SH, but more likely they will transport you to another dock for the boat and you will want to do more shore diving at their other shop down the road.
You can go to the East End and stay at Compass Point, they have something of a beach there, Ocean Frontiers is on site and their boats leave from there, but no shore diving. The Reef a few miles up the road has a beautiful beach, again no shore diving and no boats from there. There is an Ocean Frontiers "shop" at the Reef but they don't dive from there, it's more of a place to check in or buy suntan lotion. OF will pick you up for the short drive to Compass Point. There is also Morritts on the East End next door to the Reef, beach, no shore diving, Tortuga Divers on site but most don't seem to consider it nearly as good as OF.