In my experience as a lifetime DGP all Hyatt properties will handle this type of situation differentlly. But in this case, I think the hotel is sticking to Gold Passport rules. Your use of the club is a Diamond priveledge. Unless your guest is also Diamond, they should not receive the same priveledge. This is the case even if you pay for the other room.

$10 is pretty inexpensive for the club. By the way, quite a hotel, huh?