Antwort von einem DPE aus Orlando, der mir schon mal half:

"It is possible to switch the country upon which your FAA pilot certificate is based upon, however you would need to complete a new application and meeting here in the U.S. to process the paperwork as the FAA no longer allows it to be completed via Zoom.

You would need to request a new FAA Verification Letter of Foreign License from the FAA for the country you are switching to (Austria) and that would need to contain all your existing ratings. With that, it’s just a matter of filling out the application listing the new information, and then you’d be issued a FAA temporary with the new country and license number. It’s purely administrative and would not require a FAA checkride.

Let me know what questions you have about it. I’m fairly available to meet here in Orlando when you are ready to complete it."

Vielen Dank nochmals für die damalige Vermittlung, Patrick!
(Kontakt zum DPE per PM, wenn ihn jemand brauchen sollte. FSDO Orlando, nicht Washington DC.)