If the pilot cannot carry pax, that is one matter. He can still fly alone and therefore can act as PIC. The instructor is always entitled to be the commander if giving instruction or supervising (according to LASORS).
So you can conduct a flight in which the instructor is PIC and the other person is PICUS. You do the requisite flying to be able to take pax and then the instructor gets out.
Logging the above flight as PUT makes no difference, as you say Mark1. It only means your PICUS total is less than it could be.
Bose-x quotes a common view, but I think my post above demonstrates that this is not necessarily how the rules actually work...