I don't think we were quite sure whether the question was what you should carry on the bike with you as you ride or what you should take to keep at your residence.

Caribbean is pretty vague for me to guess. I've been to many of the islands in the Bahamas but not with cycling on my mind. Don't have much experience with the other parts of the Caribbean.

Double check to be certain you know what you can take in without having to pay an import duty. That can get quite pricey for some things.

If you are used to things happening right now, then take some yoga lessons and learn to meditate. Island time is a real thing down there. At least for the places I've been. So if you need to have something brought to you from the other side of the island, be sure to have plenty of beer money so you can take refuge at the local bar while you wait a day or two for it. If you don't go with the flow, you'll stress out and ruin what should otherwise be a great relaxing experience.