This may be the first time I've ever seen financially constrained and Grand Cayman in the same sentence....
The East End is going to require a car rental also due to the distance from just about everything except the few restaurants and one? food store out there. And it's about a 45min. drive back to town. Cars like everything else on Cayman aren't cheap. The Compass Point/Ocean Frontiers suggestion is a good one except they're the only operator in that area so tend to collect divers from the resort, nearby rentals - my buddy even drove over from Turtle Nest Inn in Bodden Town since it was closer than Georgetown. So possibly more of a mixed crowd than you seem to want.
Not that it's a given that you won't be with new divers but you could also look for something at the north end of Seven Mile Beach - the cruisers don't usually get that far north as Fosters in town, Red Sail and Cruisediver.net are either in town or own the cruise contracts. There's multiple dive operators who will dive the North Wall from nearby marinas - probably the 2nd best diving on GC. And almost all the condos in that area on Seven Mile Beach are oriented with waterviews. Almost every complex website shows the "view" from the property, some on vrbo.com show the view from your deck. If you did want a beach, it's a nice one in that area.
If you do it's also worth looking into dive operators in that area that cater to
smaller groups. Many go with 6-8 divers guaranteed
maximum even when the boat is rated for more. So your odds of being with other experienced divers are much greater than the "cruise ops" who can have upwards of 16-20 during busy times. CITA requires a DM for every 8 divers but I prefer not to have to wait while the other groups exit/re-board when diving. Most of them can be found here:
Cayman Islands Tourism Association - Things To Do In Cayman, Cayman Culture,Cayman Islands Watersports
My personal suggestions would be Indigo, Neptunes or possibly Living the Dream for what you want. Maybe Wall to Wall but they might pickup cruise traffic from their shop near the port. There's a dozen other advertised small group ops also.
Since you mentioned financial constraints - what about shore diving? Generally boat dives on GC run in the 2/$100+ range - buying multiple day packages gets some discount. So at 4 dives day you're probably looking at something in the range of $180PP/day of diving. Almost makes the Cayman Aggressor a better deal...LOL. We did a week on Cayman and rented a car twice. Once to drive up to Turtle Reef and once to go south. Andy's is at the airport and on SMB - they'll pick you up. There's also local bus service (small vans) running all over GC daily so you have access to shops/restaurants etc. They go by pretty often till at least dark. I don't know if you could bring gear on it though - maybe 1 bag each? The locals do bring groceries.
Several days where we did 4 tanks/per day a car would've mostly sat around - if you stay on SMB many of the diveops have van shuttles running daily and pick up is included in the price. Ours would drop us off first for lunch, drop the other divers and then pick us up again for the 1PM dives.
If you rent a car another option is to find something in the West Bay/NW Point area. The real savings in that area - or north SMB - is the shore diving. Three shore dives in that area are IMO among the best, Turtle Reef, Lighthouse Point and Cobalt Coast. Turtle Reef is on the Cracked Conch (restaurant) property with SunDIvers on site. There's a mini-wall there and the North Wall is a longer swim out. Ollen as Sundivers works with Happy Fish Divers for boat diving.
The other two locations might even be better, Cobalt Coast is the semi-AI dive resort and Lighthouse Pt. is a small condo complex. Both have Divetech on-site for diving and I've read there's now a restaurant at LP as well. Divetech does some fun dives, the West Bay Express is a scooter dive between their locations and they even offer things like rebreather tryouts. They also are in an optimal position to get to some of the more NE dives that are too far for Georgetown based shops.
Another option in that area is Bonnie's Arch condos - a small complex on the north side. There's only a few rentals in there - on vrbo.com. They have
private ladder access to the Bonnie's Arch dive site from the property. Turtle Reef and the Divetech properties are nearby also for boat diving. If there's more of you than it appears, Coconut Bay condos are another option - IIRC all are 2-3 bedrooms. About 200 yds. from Turtle Reef and they have ladder access to one of the Hepp's sites. The owner of Stingray Watersports owns a couple of those units and offered to provide tanks for us also if we boat dove with him.
Most Cayman shore dive sites are owned to the waterline by someone so you rent tanks on-site - most won't allow you to bring yours on-site or remove theirs. So for the few other sites - Smith's Cove south of Georgetown is one - rent from Eden Rock in town or Divers Supply in West Bay. Eden Rock being 500' from the cruise port you may want to do that at odd hours. In case you're planning a June-Aug trip, Devils Grotto there is spectacular when the Silversides are in. If not it's pretty barren - just you and a few hopeful Tarpon.
I agree with AggieDiver also - except for the ABC chosen. I'd pick Curacao instead since it's often cheaper to get to - many flights to Bonaire go thru there first, has the same great shore diving - with a bundled package you could boat dive mornings and shore dive afternoons for very little additional cost besides car rental. Unlike Bonaire there are condo complexes all along the coast on named dive sites - most having an on-site operator as well. Farther west or farther east of town most condos are on short cliffs above the water - a lot of the beter dive sites/beaches are in coves between them.
We rented in Lagun once and 5 of us were literally the only guests in a 12 unit "apt'. Great views, breezes but a 60' vertical drop to the water. Playa Lagun 2 blocks away has 3 condo complexes and 2 dive operators surrounding the cove/beach. Restaurants are a drive from that area though. Boat dives are about the same price on either island and there's a lot more non-diving things to do on Curacao. Either should be substantially cheaper than Grand Cayman. Diving is all small stuff on either island.