So at your club is the hirer is the commander on a club check as it does not fall into any of the categories listed above?
Thats the general idea but after all these posting on here I am beginning to doubt it myself (is the commander P1 u/s or P1.)
In your particular case, you don't have much of a choice. In your POB it says that a club check can be performed both by an FI and by a designated PPL (with a bit of additional, informal training, including RHS). As you see both of these checks as equivalent, safety-wise, it would be unfair to the hirer to be able to log P1 in one case, and Pu/t in another, just because the person in the RHS happens to have an FI ticket or not.
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.
What you want to avoid, obviously, is the situation where the hirer thinks he is Pu/t, behaves as such, logs it as such and later found out that the person in the RHS is just a PPL and not legally capable of supervising/training someone in the LHS. Particularly if something happens and people start inquiring as to who was PIC.
Designating the hirer as PIC in a clubcheck has the added psychological advantage of making the hirer aware that he is responsible for all actions, including pre- and post flight, and cannot cut corners because another PIC is on-board.