Amanda, the flyfishing part of your plan may not be so crazy after all. There are bonefish and tarpon in most the islands, and you can occasionally tempt sailfish and mahi-mahi with a fly offshore. Only the well-known spots have skilled guides.
You just need to know where to find the fish, and where you might get a work permit!
Lots of Carribean countries/islands are very restricted in who can get a workpermit, as they do not want anyone to come in and take work from the locals. So, I would look for something that does not compete with local talent, or something that will employ a few locals. Will make it much easier to get the necessary permits.