I'm assuming that shouldn't be a problem.

PC benefits are going to be tied to your Hertz Gold account. While the status may have been given because of the card, once it's confirmed, that should essentially be as good as having the status from meeting rental criteria. You can choose to use whatever CDP code makes the most sense, and there isn't any restriction that I am aware of where the Amex CDP requires using an amex to book (although presumably if booking via Amex travel, you can't use a non-Amex card, but should be fine if booking directly with Hertz or another TA/OTA).