I've been cruising and diving for 12 years, starting as an OW diver. How much gear to bring depends on several things. First is how comfortable you are in using different gear. If you are not comfortable, bring all of your own gear....last thing you need in an emergency is fumbling with unfamiliar equipment. Regardless of this I will not travel without my own computer and regulator (and of course mask, fins, snorkel) Second question is how much are the shops charging for their rental gear. Western Caribbean tends to charge for each piece, while many of the Eastern Caribbean only charge $10 for everything. Not that big of a deal with one diver, but my whole family dives so I can get into 80-100 per trip for renting a BC and wetsuit. Next: What kind of room do you have? We always get a balcony room primarily so we can put our gear outside between dives after rinsing in the shower. An ocean view or interior room can get pretty ripe by the end of the cruise.
So all of this adds up to basically three scenarios. 1. Take minimal gear...mask, fins etc, plus computer, regulator and basic tools/save a dive). We packed the fins and boots in a checked bag and carried the regs, computers and masks on the boat in a pull behind. 2. Add your BC and rent wetsuits. Huge benefit here is the clean up....the one item that will weigh you down and also smell up your cabin is the wet suit. We added the BC to the bag with fins and checked it. 3. Bring all of your own gear. This is the most comfortable for diving because you already know exactly what weight you will need, it fits right and you know where every thing is, BUT its a lot of gear to schlep on and off of the boat every port. For the ones you are going on the ports are all well maintained and easy to navigate, so not as big of a deal. Some place like Grand Cayman where you tender in (little boat to shore) that can be a real hassle.
One other option we usually take for our sensitive gear is to pack it all in a tote. We use an 18x24x18 tote, put the regs, drybag with computers, cameras and masks in it. I drilled a hole in each end and put a master lock on one end and a TSA lock on the other. Works great and the baggage guys can't crush your important stuff.
Last thing to think about is your accessory gear. If you choose to rent a BC, make sure to pull off your whistle and See Me Buoy. I usually leave my knives at home, but leave my shears in place. The shears are more functional for safety and the security guys don't make you turn them in every time you get on the boat.
Now for the dive locations. I went on this exact cruise in Dec, but I didn't make my dive plans early enough and got shut out of diving. I would recommend booking with the cruise line right now while you work your own plans, that way you are guaranteed a dive in each port (only time I've ever used the cruise line is Belize). Some friends on the ship dove all 3 locations and enjoyed all. The only one they used the cruise line for was Tortola. We did a snorkel trip to the Rhone and they dove on it. It was a good dive but not worth two tanks. The shallow end is about 30-40 feet and I free dove down and got some great pictures. In Antiqua they dove with Extreme Scuba and would recommend them. Nice operation. The most recommended operations in the other locations (scuba board and travel advisor) were Admiralty Dive Center and blue Island Dive Center in St Thomas and Dive BVI and Sail Caribbean Divers in Tortola.
Hope this all helps! Enjoy!