As a past food establishment owner on the island, I will give you a tip.
DO NOT EAT any "wood fired" pizza on the island. There is no wood grown on the island worthy of cooking. I do know for a fact that most wood that has been used was pallet wood from shipments coming on island. This wood is heavily treated with pesticides, and chemical wood preservatives and is harmful. Proper cooking wood is too expensive to ship. I looked into it myself and that was the only alternative, so I would advise highly against this.