Grand Cayman will provide a beach as well as good diving, yet it will also offer more expense. The Brac will offer a lot more quiet with yet better diving than Grand and will be the least expensive but you have one choice for an op, Reef Divers (Brac Reef Resort). The best diving would be Little Cayman of the 3, and expensive as well, yet, the last two do not offer much of a beach if that is what you have in mind.
Turks and Caicos would be the most beautiful beach in the Caribbean and with Fifi (Caicos Adventures) you would dive off from west Caicos (a wonderful 45 min off the coast of Provo).
Grand Turk has some beach as well..not as beautiful as Grace Bay yet good enough, wall diving is on the menu which I love yet Grand Turk is a harder place to get to.
Bonaire is shore dive heaven, not much beach available on this Island either but the diving is great with the small stuff especially. Watch out for the cruise ships (pod people) while in town.
Dive at your own pace and when you want. With a car rental the price does add up as well as getting to Bonaire.
Cozumel, can't be beat for price and some quality dives, yet you have to think about drinking the water and what they washed the lettuce with when having a salad. I guess I wouldn't risk my honeymoon on it.
Hawaii, this would be a top choice, but it's not the Caribbean.
I like the idea someone stated about a liveaboard in the V.I.'s, how about the Cuan Law, or in the Bahamas, Aqua Cat? We've done several liveaboards but none in the Caribbean.