Shouldn't a simple search of the relevant state's bar registry be sufficient to realize the lawyer is fake?

I guess you'd have to at least suspect the letter was fake to try that, so you might miss it if that's not even on your radar.

Depends on the region. I usually write (or now have gpt write) a letter with a bunch of fines in it back and that almost always shuts them up. I have never complied with any legal actions against me; I have been to court and that was cheap without a lawyer as well. I did nothing wrong, so I am quite confident I will always win; so far going strong for over 30 years since I am an adult and received such nonsense directed at me or my company.

I would never live in a country (guess US) where you need a kidney to defend yourself even if you did nothing wrong.

Or you can just wait it out.

If they are serious, they will follow it up - and you will eventually have to show up in court.