I do have an upcoming cruise scheduled for November with stops in Cozumel, Belize and Grand Cayman (GC is on the itinerary, but likely won't happen as the island looks to remain closed to cruise passengers until 2022). In Cozumel, I'll likely be using one of the local shops, but in Belize I've already booked a ship excursion. Belize is a tender port, and for the reasons mentioned above, I figure it'll be worth the extra $40 or so to avoid any issues getting off the ship.
I've dove from a cruise ship multiple times at all three ports. My two cents:
Cozumel: more dive ops than you can count, and the marina that most of them leave from is just down the road from the cruise terminal. Find your favorite dive op, contact them and make sure they can take you diving with your ship's schedule (some dive shops like to get their dive boats out early, before the cruise ships let their passengers off.)
Grand Cayman: if you do somehow get the chance to stop here... as you noted, this is a tender port. Makes getting off the ship a bit less predictable, but the dive shops on GC will work with you. Do your research, communicate with the shop, find out what they can do. I've done this and have always been able to work it out.
Belize: assuming your stop is in Belize City... this is arguably the worst cruise stop in the Caribbean for diving. Not because of the reef or dive sites, but simply because of the lack of options. It's a tender port, but I've found only a couple dive shops in Belize City. The one I used (only one time) had the worst customer service of any dive shop I've ever been to. I assumed it was because they had no competition, so they didn't care what customers thought of them. Even worse, the area with the best dive sites is beyond where the cruise ships anchor. So we would take the tender from ship to shore, take the dive boat back out from shore to the area of the ship, then back to shore, then the tender from shore back to ship. A lot of wasted time.
One cruise fixed this problem: for the ship's dive excursion, the dive boat picked us up directly from the ship... using the tender platform. We took off directly from the ship and headed east to Turniffe Atoll. At the end, the dive boat brought us directly back to the ship. It was a little more expensive than going with the independent dive shop in Belize City, but more than worth the extra cost.
When booking your dives at your Belize City stop, get the details on the ship's excursion. It's the only one I've ever booked through the ship.