The government being this sloppy at getting accents right is surprising, I would expect them to value accuracy and a clean paper trail when handling names.
That tells me you're German, I didn't even need to see the ä and ß.
Even in the UK I encounter websites that won't accept my Norwegian address because it begins with Å. English speaking countries generally are pretty bad at this sort of thing.
My Spanish girlfriend has an ñ in her last name, and as does our son. To the people here in Norway, I just tell them to put a plain n when typing the last name. It’s easier to just go with that than to try and get people to understand how to type ñ on the keyboard (even though our computers can do it), and to avoid extra back and forth with people who have systems that don’t handle it.
Likewise, when I’m in Spain I don’t bother to say that my last name has ø in it. I don’t even bother to rewrite the o in my last name as oe. I just put it as o.
The only situation where I put it as oe is indirectly when an airline converts ø to oe on my airline ticket, or where the airline system doesn’t handle ø and I put it as oe for them when making the booking. To me my name looks worse with oe in it, and seems harder to pronounce for people if I write it as having oe in it than just putting it as o.