When did you try to get tested? I booked a test in June 2020 and got it done the same day. Same thing this spring, I started feeling symptoms around lunch and had no problem getting a test done in the afternoon.
Tutanota unfortunately has the same drawback in that you have to go to the settings and register a new email alias before you can reply from that adress.
Presumably it's a client support thing, as in if you use a third-party client with Fastmail/Tutanova that does support it, they won't mangle the headers?
I'll get around to testing it I suppose, just not something I thought of or realised I'd lose when I switched away from Gmail.