You won't have time for a boat dive in that window - it's 15mins or so each direction to the reef/wall and the dive takes 45mins+. The better dives which most of your fellow divers are paying for are even farther to the north. Add in gear-up/diver recovery time and it's so tight that I wouldn't chance it. Plus you won't be diving with a cruise excursion in that short window so your ship
will leave without you if something goes wrong. Most of the afternoon dives also leave around 1PM.
A private charter where you specify the time is going to be about $6-800?. Deep Blue or Wall to Wall could
possibly do that with advance notice - they're just north of the cruise port at Lobster Pot. Maybe Off the Wall also - same place. Don Fosters to the south is the cruise operator for many cruise lines. All are listed here:
Cayman Islands Tourism Association - Things To Do In Cayman, Cayman Culture,Cayman Islands Watersports Kittiwake is also by appt. so you'd have to find out which shop is going there on your day.
Two options you could do - Eden Rock is 500' from the cruise port. It's fairly popular, not the best on Cayman since a lot of cruise divers have dove it over the decade or so that cruise ships have been calling. No guarantees but you might be able to find a buddy there - or maybe on your ship.
Cayman Diving Guide The silversides shown on the website - that's summer only. The Grotto is still a decent group of swim-thru's but it will be mostly you and some optimistic tarpon hovering in the shadows.
You could likely dive at Sunset House - it's a few mins. by cab in south Georgetown. Almost walkable but not on a tight timeframe like yours. There should reliably be groups shore diving there in the afternoon you could ask to join. My Bar there overlooks one entry and most people sitting in it are divers. At noon though eating lunch - their afternoon boat dives to Stingray City or the Kittiwake leave at 1PM. They're too far south to get you there and back in time.
There's another entry a little north at the Salt Water Pool so if you see dry geared up divers walking that way - they're probably headed there. From Sunset House you will be able to see your ship.
Sunset House - George Town - Cayman Islands
You probably won't have time for the Kittiwake even if you find someone within your window. It's off the far north end of Seven Mile Beach so probably a 1/2 hr. each way. It's a shallow dive for OW divers on the boat so they all probably manage longer dives.