Yes hard to sue the government in general, and it typically only pans out when they are in aggregious violation of law or common sense.

FAA aeromedical taking their slow bureaucratic time to address a certification issue is well within the scope of their authority, I think you'd have to show malicious intent to have any shot at a liability claim, ie somebody didn't like you personally and deliberately stalled the process.

I've also never heard of the FAA, or any government agency, being held liable for lost wages due to slow processes. I'm sure the precedent is already set against that, otherwise people would do it all the time (or the FAA would process cases quickly to avoid it).