I don't think, Step, that I'd be willing to strap into the right hand seat under the circumstances you describe.
The instructor you are talking about is there because the flying standards of the chap in the left seat are not 100% certain, and because the instructor is a known higher ability pilot who is making a decision about the outcome of the flight. So for them to have the ability - legally - to take control seems essential. And if they have that, they are Captain.
The person in the left hand seat may in some regulatory regimes be P1/S, but the person in the right hand seat is the "S".
G