I agree that FHR could be a huge benefit, assuming OP could take advantage of the benefits. I have never seen base rates lower through FHR for FS properties. The times that I have noticed, the base rates are a few dollars more through FHR, though when factoring in the benefits, the FHR deal is much better.

And while the terms of the FHR program purport to require payment via amex, I have been able to settle my tab on my CSP. There is an entire thread around here somewhere on the topic. The consensus appears to be that most hotels don't seem to care how you pay, as long as they get paid.

I assume that most (all?) FS properties are available through FHR, but I haven't actually checked.

Depending on how much OP plans to spend, he may want to consider the Plat for access to the FHR program, with the actual spend put on a CSP (assuming he's comfortable with violating the technical T&Cs, and assuming the property doesn't pre-charge the stay on the Plat card anyway -- I can't say I have actually ever stayed at a FS via FHR, so I don't know their practice).

Either way, I wish I had OP's problem myself.