OK, I just disabled safe-mode for non-admins.
For now, I don’t need absolute security in blocking the name change. The people who are playing with the rules are not technically savvy enough to even know what the development tools in their browser is made for, and they have absolutely no idea what html or CSS are. So, to me, the CSS method is a great initial solution.
If I can make it still a bit more difficult next week by adding this as a plug-in, then the security level I have is more than enough for my current needs. In any case, my forum is fully closed to non-members, so in the worst case, I can even have a disciplinary measure voted by the board and ban them from the forum. I just want that the option to change their name is not tantalizingly present on their profile.
And yes, I have a manual approval that is performed after I have checked that the members are valid and the forum is by invitation only.