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.
Ü isn't even a special character or utf-8 - ü is part of ascii. How does this even fail? Is their database a 7-bit database?
But I wouldn’t bother memorising that and every other possible way that the other person has to press the keys depending on their keyboard layout and operating system. I’d just tell people to put u instead.
But that’s not really the point. No matter how many keyboard shortcuts the clerk at the DMV memorizes there is always going to be some text that they just cannot reproduce accurately. Whether it’s an accented character from the exotic land of Spain or some real Zalgo, something is going to get lost. No individual human can correctly deal with all possible textual forms.