Flying Lawyer you have opened the big can of worms of who is PIC.
As Commander Chuck Street hasn't got a CFI licence who is??? (don't believe he has)
You are flying a flight for hire but it's only Part 91. But if you only have a UK Private can you legally do the flying???
As all PIC is flown in the left seat then who is PIC???
UK regulations require all PIC time to be flown from the commanders seat. I will admit that in the US that isn't the case.
If he has a CFI and you're PIC from the right then all I have written is wrong. Except the Part that he is getting paid from the flight.
I'm sure you will tell me that its all legal but as a qualified US CFI we never recommended it to the European students.
Heard that the UK used to delete time flown on the news flights from total time for a Commercial licence application.