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.

I think there is a belief (right or wrong) among a lot of lay-people and non-lawyers that the legal system is an opaque minefield of "gotchas" where if you don't file form 27B/6 within 30 days, with a specific magical incantation X written on it, and then appear before a judge and speak magical incantation Y, then gotcha! you lose! So any time something looks even remotely official, people panic and wonder which minefield they are going to inadvertently step on, even by ignoring it.