Hi,
I would recommend you ask this in the BizJet forum where someone more clued up with these matters might be able to help you. Though my hunch is a moderator will move it back here for the sake of conformity over a genuine desire to help you (because that's how some of them roll!)
In case you don't get any help... I'll throw in a few lines.
The Caribbean islands and Central American countries generally align with the US (FAA). Having an N reg based there is not going to be a problem but make sure your story has rotations built in just to be sure. A pilot flying an N reg aircraft needs to have an FAA license. EASA pilots cannot fly N reg aircraft (there are some very special authorizations concerning delivery/ferry flights. An area I don't know much about).
All the best.