There is a long-established "right of innocent passage" from the UNCLOS that allows doing just this, with certain restrictions. I'm not going to hot-link either keyword, as you can
google the terms and choose which of the
references to use.
I have used this "right" numerous times in the
Caribbean, dropping
anchor at night and not going ashore and departing the next morning. I usually have a reason prepared should I get asked ("I was so tired I thought I'd pass out", "The
engine sounded strange, so I stopped to check it out", "my crewmember was so sick that I thought I'd need a doctor", etc.) to explain my use of this this UNCLOS right, to which the Caribbean nations are, as far as I know, all signatories.
The U.S. Islands are notable exceptions, I would not dream of trying to assert my UNCLOS rights as those authorities are exceptionally strict. They will usually track you once inside territorial waters. The other Caribbean nations are more lenient or lax, and I've never once had any issues.
But I would recommend on stopping, clearing in, and enjoying the wonderful islands and nations you will traverse on your trip down to Grenada!
Your first stop will most certainly be more than just an overnight - you'll be happy to step ashore after the
passage from the
USA and will most likely need to reprovision and perhaps buy
parts for items that broke on the passage.