Generally speaking, I’d say no, this isn’t how Hyatt or Privé is designed to work. Privé bookings need to be created by the advisor from the outset, and an existing direct booking can’t formally be converted, nor can an SUA be carried over into a Privé reservation.

That said, every case is a bit different and ultimately it comes down to the property and the specific rate. In some instances, a hotel may choose to extend certain Privé-like courtesies. If that were to happen, it would be at the hotel’s discretion and wouldn’t require rebooking, so you wouldn’t lose the SUA. But this is the exception rather than the rule and not something an advisor can guarantee.