Eden Rock downtown rents tanks for off-site use as does Divers Supply in West Bay. If you're diving a site with a shop on-site - rent from them. They don't allow/discourage outside tanks since it's their property to the water's edge. Also most have added docks/facilities etc - including the all important ladders over the ironshore. It's not the ironshore, it's the drop...lol.
You're too early for the Silversides migration so pass on making Eden Rock/Devils Grotto one of your shore dives - lots of cruisers haven't done either much good. Both are pretty average sites - the Grotto has a lot of rocky swim-thru's.
I'm a fan of Turtle Reef having
never dove Lighthouse Pt. Head west and you'll run into a Tarpon cave at about 40' along the mini-wall. I'd dive both since you're going to be on-site at Divetech - Turtle Reef is a 5-10min. drive east. Ollen at Sundivers used to provide guide services thru Happy Fish Divers at night if you want one. It's pretty hard to lose track - there's not a lot that looks like Macabuca/Cracked Conch from the water in that area.
I'm guessing Divetech still has their scooters so they might do some fun escorted dives with those also. We actually rented them w/o a guide (and little to no scooter experience at the time) but that may have changed. They used to run the West Bay Express between Cobalt Coast and Lighthouse Point when they were on-site at both - they follow with a boat.
If you don't want to drive all the way north for an afternoon shore dive - TNI is reasonably close to Sunset House, the AI Dive resort in Georgetown. Fun afternoon: dive the Mermaid, the Nicholson (small) wreck, Cathy Church photo gallery/store and My Bar has good food.
Smith's Cove is just south of there - it's a shallow dive but you'll have to get tanks in Georgetown first - Sunset House has a no removal policy. I thought Divetech did also but I'm not sure.
The shops/ops I mentioned:
Cayman Islands Tourism Association - Watersports
Some of this is pretty dated:
Scuba Shore Diving Site Listing for: Cayman Islands, Caribbean
My buddy stays at TNI and dives with Ocean Frontiers. Ask about Babylon among other dives. It's actually a shore dive also but I think you'd have to get tanks in Georegetown - I've not read that OF, Red Sail/Tortuga Divers (at Morritt's) or the operator at the Reef Resort rent offsite.