The person who is PIC needs to be legal in the aircraft.
Assuming the person receiving the BFR is legal as PIC there is no reason why he should not carry a passenger who happens to be an FAA CFI.
Which rather brings it back to what the FAR's say about it as to whether the BFR can legally be signed off in the circumstances.
Whether or not the pilot can pay someone to be a passenger is an interesting question. It is probably covered by the phrase
Subject to the provisions of this article, aerial work means any purpose (other than public transport) for which an aircraft is flown if valuable consideration is given or promised in respect of the flight or the purpose of the flight.
It wouldn't be too difficult to argue that the purpose of the flight is the BFR and therefore any payment to the CFI is payment for the purpose of the flight, which would turn it into Aerial Work and require the PIC to have a commercial license.
Article 130 of the Air Navigation Order available
here is probably the reference you need.