This may not be helpful since it is not in the Virgin Islands, but I rented a place like you describe in Islamorada FL Keys. It was right in the middle of the Key on the Atlantic side and was in the middle of the downwinder that people run there between Whale Harbor and Anne's Beach. There are miles of knee to waste deep water (tide dependent), and the house provided me with my own private beach with just enough room to keep 3 kites inflated by the house and easily self launch. It couldn't get much better for kiting. I will PM you the address if you are interested.
Best of luck with your search. I am interested to hear if you find such a place in the VI. That would be worth the trip.