You can't have it both ways, unless you want to have this discussion all over again each time a club check is being performed. So if the POB specifies that a designated PPL can perform club checks, then in all cases the hirer should be P1, even if the guy in the RHS happens to be an FI. And if the POB specifies that clubchecks can only be performed by FIs (as my club does), you can (but do not have to) specify that the FI is P1, and the hirer Pu/t.
Experience widens the mind though you see BP.
Around 30 years ago I worked at a flying school that had a very poor reputation, I arrived one morning for instance to see a F27 mainwheel being loaded into a Grummnan AA5 to be ferried to another airfield. I could write a book about all the incidents that took place in the short 6 months I was there. The CAA had a major interest in the school and visited on a regular basis, in fact I saw more CAA inspectors in that 6 months than I have seen in the rest of my flying career.
Before I got to the school they had employed an instructor who they noticed was very reluctant to fly with students who had not gone first solo, making excuses about the weather etc. After two weeks of this they found out that not only did their new instructor not have an instructors rating, he did not have a pilots licence either, he was however a licensed engineer.
Now you tell me who was pilot in command on those flights and how were they logged?