Mike..nothing wrong with a Cat36 in good shape for what you are proposing. I assume you would plan on taking her down the thorny path rather than the ocean route. The main advantage of this route is the lack of anything more than 24 hours at sea before you can duck in to a harbor. The disadvantage is that you face about 700 miles of motoring into trade winds and seas between the Turks & Caicos and the Virgins. We can talk about that another time...but I'd encourage you to think about expanding your fuel capacity and rigging a fuel system that can be easily attended to at sea (dual racors). The other thing that is best done in the states is rigging an SSB which is really a necessity as you head south of the Bahamas. I will assume you'll prepare the boat for heavy duty anchoring tasks and appropriate safety precautions (jacklines, liferaft,Epirb) but there are lots of boats of "coastal" construction that have made that trip safely. The only other thing I would add is that you CANNOT be on any kind of timetable for the passage or the boat will get killed! We had to wait for 30 days straight in the DR for anything LESS than 20 knots on the nose and 6-8' seas....and several times for a week or more in other spots.
Go for it Mike!