Reading the source code, Flarum is supposed to show the Display Name in mentions https://github.com/flarum/mentions/blob/master/src/ConfigureMentions.php
And you're also supposed to be able to search via Display Name when mentioning https://github.com/flarum/mentions/blob/master/js/src/forum/addComposerAutocomplete.js#L100
So basically insertion should work just like in Discord, where you can press @
, type any nickname you know the user by, and it will insert their actual Discord username even if it's completely different.
In addition, the mention block should show the Display Name in read mode, so discussion readers shouldn't even see the username that was typed in the message.
I have however not tested the Nickname extension. If the above isn't working, consider reporting it as a bug in Flarum.