I've transferred onboard booking from RCI to another agent, and had absolutely no problems at all. I would think that transferring the Princess On Board Booking to Best Price Cruises would work the same way. I have used Best Price Cruises in the past as well, and have had absolutely no problems at all. They seem to be a very reputable and reliable agent.
Have a great cruise,
Marine, Pilot, Singer, Tuba Player, Cruiser!