There actually is two dive operators a short drive away. Tortuga Divers/Red Sail has a location at both Rum Point and Morritt's resort. Ocean Frontiers is on the south side of the East End so a longer drive. My buddy/b-i-l liked OF a lot earlier this summer. All my diving has been on the west side so I only know of them. There's also a third operator at the Reef Resort but since Morritt's is basically next door IDK which is a better choice.
If you're planning to drive all the way around to the Seven Mile Beach area - go 5 miles farther north and dive Turtle Reef, Lighthouse Point or Cobalt Coast. IMO Turtle Reef is the best. Head south and look for the Tarpon Cave - it's pretty cool to move through them. Also watch overhead - we often saw Eagle Rays flying overhead there. My pick for west side operator is going to be Wall to Wall most of the time but I know the owner. Neptune's Divers seems to do a good job also and Ambassador gets some good recommendations. All based along SMB. But there's really a bunch of others -
Cayman Islands Tourism Association - Things To Do In Cayman, Cayman Culture,Cayman Islands Watersports They all charge roughly the same so bad operators don't last long on Cayman.
If you wanted to make a day of it book a Kittiwake Dive with one of the west side operators - or either of the two I mentioned ^ also do it - I believe OF asks you to meet them at a west side marina though. Also there's Stingray City dives - boats for that leave from the west side of the north sound. Just about every dive op offers that dive.
In south Georgetown Sunset House is decent also - there's the famous Mermaid, a small landing craft wreck, the Cathy Church Photo Gallery/Store and My Bar had good food and will be all divers. It's far enough south to not be affected by the 4-5 cruise ships in port almost daily.
The only places I know to rent tanks are on the west side. If your plan is to shore dive near where you're staying you might see if you can arrive early enough to pick them up b4 you drive out. Eden Rock downtown and Divers Supply in West Bay are the two options. Also the shore dives I mentioned above all have dive operators on-site. I do know for a fact that Ocean Frontiers does not let tanks off-propterty - no idea on Red Sail/Tortuga.
If you're feeling ambitious Babylon is shore accessible. But it's a surface swim and not the easiest/most marked entry point. Also there can be some boat traffic. It's arguably one of the top 5 Grand Cayman dives though - I believe Tortuga does it almost daily also via boat. Even some of the west side operators ride over for it on extended trips.
My other must dives:
Ghost Mountain
Trinity Caves
Big Tunnel
Orange Canyon
Turtle Reef
If you're boat diving on the west side and the operator asks - Aquarium is a good shallow 2nd dive. Watch for fish hovering around the coral heads - those are cleaner stations. Hold out your hand and the shrimp might clean your cuticles.
hth,
P.S. - something your rental agent may have not mentioned - the food stores on Cayman are all closed on Sunday. There's a Fosters market just as you leave the airport - open till 11PM Sat. night because of it. So you might want to stop for a few things. It's going to be the better part of an hour drive so plan for that also. There's also a small Fosters (Hurleys?) on the East End - IDK any details about it.