Most any of the tour operators will do private charters, if you are willing to pay for it. I talked to Seaduced about this the other day for a honemoon couple.
Now, for the secluded island: used to be this wasn't hard to find but is doable if you chose the right day. The cruise ships are visiting many of the undeveloped islands (especially one of my old favorites - Rendevous Caye).
Also, as Peter said, you could try north Ambergris but no guarantee that other humans may have the same idea on the same day.