All sorts of people from non-English extraction will have the same problem.
I've known a few people called Zoë (enough now that I still know the alt-0235 short-code for it on windows), and inevitably on their licenses the omit the diaeresis above their name.
It's really a version of Falsehoods Programmers Believe about Names[1] writ-large, by government.
I was curious and googled a bit. Some British people on netmums.com report that they successfully put these names on birth certificates: Éilis, Zoë. I would guess that "Michèle" would also be allowed. I've never heard of an official set of allowed characters, but everything relating to names of people seems to be unregulated in the UK.
One argument people mentioned (though I have no idea whether it has any validity) is that if you have the diacritic on the birth certificate then it's easy to drop it later, or not use it in practice, but the other direction would be harder to justify.
but everything relating to names of people seems to be unregulated in the UK
As an aside, one can indeed simply change one's name whenever one likes and start going by a new name, so long as it's not for fradulent purposes. One is advised to get some kind of documentary name-changing paperwork, and getting a passport etc in your new name will require that paperwork.
It is however slowly going the way of the þorn, since most people don't know how to type it, and even if they did they wouldn't see a reason to use it.
I've known a few people called Zoë (enough now that I still know the alt-0235 short-code for it on windows), and inevitably on their licenses the omit the diaeresis above their name.
It's really a version of Falsehoods Programmers Believe about Names[1] writ-large, by government.
[1] https://www.kalzumeus.com/2010/06/17/falsehoods-programmers-...