Signed letter is not equivalent to TM30 in any shape or form. What you need is a screenshot from the online registration system.
Did you check in more than a week ago? That stupid online system only allows you to go back one week or so. They probably reported your stay as required, but can’t produce evidence
The easiest way to do it is to fire you and switch to contract-based employment. I know lots of people working for multinational companies headquartered in Germany and everyone based outside Germany (EU, US, Ukraine, Russia, Brazil, you name it) are on "freelancer"/"consultant" contracts (working full time, some people have been working like that for more than a decade)
Well, it was similar to opening a bank account - first they ask for a WP, then dig through their internal instructions which don’t mention TE, and then you get “mai dai”
It doesn’t matter if there’s a regional office or not. It’s a separate entity, period. As long as you’re not paid through that regional office, you’re fine