Heyo.
So making sure I understand the situation correctly:
You have an NA account that you would like to move to the EU server.
You also have an EU account currently.
You want to name change your EU account so that when your NA account moves over, it retains the name that would've been a duplicate and otherwise changed.
Your NA account also holds the same email as your EU account.
To avoid the email issue, you intended to change it before the move, however are having issues with the 2FA
If this is the correct situation (sorry your text was a little vague what the specific issue was)
Here are my suggestions for you:
We don't grant name changes for no reason however you are able to use doubloons to change your EU acc name via the Account Management page on the website.
If you do not want to do this, then you will need to change your NA account name after the move via Customer Support (CS)
If you are having issues with your phone linking you have a few options:
Unlink your phone for now through account management.
Contact support who will be able to remove it for you if you are not receiving the texts (personally I get my 2FA codes usually within a minute)