You probably don't want to book with Red Sail then, they have large boats and limit bottom time by using 72's.
A lot of the Cayman diveops pick up all along Seven Mile Beach. Some of them will also pick you up just a short walk from the cruise port. I don't know which ones anymore but several that are recommended here often are: Neptune's Divers, Ambassador Divers, Off the Wall, Wall to Wall and Indigo Divers. And there's a couple dozen more. Most of the diveops can be found here:
Cayman Islands - Grand Cayman, Cayman Brac and Little Cayman, Travel and Tourism Site
Don Fosters is just south of the port within walking distance - they do the bulk of the cruise traffic though. I'm not a big fan of big boats with lots of divers on them so no comment other than that.
Also it's theoretically possible to do 2 AM dives followed by a Stingray City dive if you're there all day - and get off the boat early enough. Unless things have changed recently, Georgetown is a tender port, the ships all moor out in the harbor.
You'll be driven to the sound and board a boat for the short ride to SRC. It's really a lot of fun getting mauled by the Stingrays. There's a dive and also a snorkel, the dive is in 15' for about 45mins. the snorkel you're at the Sandbar nearby and can stand up. Non-divers can also snorkel on the dive with some ops. We had a family on our trip, the parents dove while the kids snorkeled around us.
Dives to do on the west side of the island: Trinity Caves, Big Tunnel, Orange Canyon.
If your plan is to shoredive:
I thought Eden Rock was very tired - lots of junk dropped from ships in the harbor, not very deep - all we hit was about 40' - a cruise ship will be sitting just offshore there all day so it takes a beating being walking distance from the cruise port. And in some places a long swim over sand between coral formations. The Grotto (same location) is supposed to be more interesting, some overheads etc.
Sunset House is a 2min cab ride away in South Georgetown. It'a an AI resort but you can shoredive there, boat dive with them, get lunch at My Bar - the divers spot and tour the Cathy Church Photo Gallery in the main house. The Amphitrite Mermaid is there in 55'