A number of our neighbors have brought their dogs. One brought his chocolate lab, Godiva, every year. Some have hitched a ride on private or corporate jets - lucky them! A charter from Florida must cost plenty.
I wouldn’t trust a dog with a connecting flight, but maybe on a direct flight in winter. So if you could get a direct flight to San Juan or St. Thomas, then charter a flight to Beef, you should be okay. I’ve seen quite a few dogs in San Juan airport post-Irma.