This isn't the FAA, or even TSA directly.
This is the local municipality, they control your airport (and probably own it). They have to do certain things to comply with TSA requirements, but the details are up to them.
The FAA guards their regulatory pre-emption with regards to aviation rules, but the fact that you have to pay a fee to get a badge to access the local airport has nothing to do with aviation regs. If they outright banned all GA pilots, then the FAA would take an interest.