I can't advise you on Caye Chapel but if I were you I would contact them via their website. I believe they're helpful. Of course, that island is dedicated to the golf course that occupies it all, and you'll probably identify more with them if you want to play a round or two!
Other than BYC there are no other marinas in San Pedro, nor is there any other dock you'll be able to get in to. There isn't one at Caye Caulker, and you've already been told about Cucumber Beach Marina just south of Belize City. But there are lots of places just off San Pedro you can safely navigate to (with help) and drop anchor. Won't help the pooch, but you can tender in and tie up in lots of docks, so long as you don't stay there overnight. You can easily tender fresh water and supplies, though obviously you'll need suitable containers. You must be careful where you take the big boat to, as there are many areas inside the barrier reef with insufficient draft.