Most operators are used to dealing with new/cruise divers there. So it shouldn't be a problem.
If you wanted to pick a couple with a higher % of newer divers that would be Red Sail or Don Fosters. Fosters used to do almost all the cruise dives but they recently gave up the contract. Downside with them is that you'll be part of a larger group of divers in any skill category - that's how they assign DM's/guides. So maybe do that once or twice then pick a smaller operator for the rest of the week.
But really any of the SMB-based operators can accommodate you and more experienced divers at the same time - they're used to splitting groups and since it's the law, they'll assign a DM/guide to each group. Most DM's on Cayman are actually certified instructors, demand for jobs there allows them to be selective.
Many 1st dives off Cayman go deep of the wall. It doesn't mean you have to. Just tell whoever you select about your experience level when you contact them. Most 2nd dives on Grand Cayman are going to be above 60'. And most moorings at the deeper sites are at that depth or shallower. At most there's plenty to see without needing to go any deeper.
One of the signature dives - Stingray City is 15' deep. And it's timed at 45mins. - if you run out of air sooner, surface and wait on the boat with the captain. I once saw someone swim up, drop their gear and then snorkel back down. But it's likely you'll have enough air for most/all of it.
If you want to try a wreck
later in the week, the Kittiwake is an easy one. It was dive-prepped before sinking in 60' - the top almost breaks the surface. You'll only be allowed on the first two (or 3?) levels and it's required that the operator provide a guide for people in your group. On a smaller boat - that could be just you - if the others have AOW, they're allowed deeper - many dive operators only take 6-8 divers max per trip.
Here's a link to the
Kittiwake website. There's also a list of several dozen CITA operators down the right side of the main page. I often recommend Ambassador Divers, Neptune Divers, Wall to Wall or Indigo Divers but honestly they're all good - it's a very competitive market and everyone charges the same so the differentiation is that they all provide good service. Plus prices are pretty much the same anywhere also.
If you wanted to do some escorted shore diving, Eden Rock is a really shallow site and with a little notice they can provide a guide. Try to do it when the cruise ships aren't in as they're about 100 yds. from the cruise port. Since you're staying two weeks, Saturday would be good. Or check the schedule
here. The Grotto is the better part of that dive - it's shallow and has a lot of swim-thru's - most open to the surface so it's relatively benign for newer divers. You will see some Tarpon lurking there - 3-4' skinny silver fish. (they're not barracuda...lol)
Turtle Reef on the north side is about the best shore dive that's easily accessible, it's shallow, you walk in off stairs, and Happy Fish Divers can provide paid guides there thru the on-site operator - Sundivers. Off the mini-wall there you'll see lots of turtles, tarpon and occasionally rays. The mini-wall bottoms out in the sand around 45-50' going south and 60-70' going north so it's not a very difficult dive.
The only exception would be if you're staying diving on/near the East End/Rum Point/Old Man Bay area. Then your choices are either Ocean Frontiers at Compass Point or Tortuga Divers(Red Sail) at Morritt's - it's all that's out there. My buddy had good dives with OF last January.
Videos for many of the dive sites:
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