All of the diving in Belize is easy, even the Blue Hole, so no issues there.
Based on your wish list, I recommend Hamanasi in Hopkins. It's fairly upscale with its own dive operation in-house. You and your brother could spend some time in Hopkins or do other mainland stuff, like cave rafting, zip lining and/or Mayan ruins clambering. You would even be fairly close to Actun Tunichil Muknal (ATM), which is a once in a lifetime type experience.
You probably wouldn't be able to book an entire boat, though. If that's at the top of your priority list you might want to talk to Ralph and Patty at Splash in Placencia. They have four boats, a couple of which are pretty small and depending on how booked they are they might be able to accommodate you. I know that under the circumstances they would try. You could also try Turtle Inn and Roberts Grove in Placencia. Definitely something you would need to arrange in advance. You didn't say when you plan to go but your chances of scoring your own boat would be better in the summer, during the slow season.
If you're willing to skip the immersion into Belize, then there are several resorts on the cayes that are exceptional: Turneffe Island Resort, Turnfeffe Flats and Blackbird Caye to the north and Hatchet Caye to the south. They're like summer camp for adults in that all you do is eat, sleep, dive and relax and everything's within a five minute walk. You definitely wouldn't be able to reserve your own boat, but the number of divers would be small, especially in the summer. When we stayed at TIR twice, we never had more than six divers and sometimes as small as two or three on each of their two boats.
You could also consider splitting your time between one of the caye resorts and spending time on the mainland. If you did that, then I'd recommend heading into the Caye District. Lots of "eco-resorts" in the jungle around San Ignacio, and you would be really close to ATM and the Caracol and Tikal ruins. Or go really remote and stay at Chan Chich near Gallon Jug and check out Barry Bowen's estate and still be relatively close to those three.
Skip Belize City unless you need to stay there to get out to one of the cayes. It's a rough town, to put it mildly, and not really set up for tourists.
Good for you for taking your brother on a trip under these circumstances. Lotta love there.