[lmm]

Your example fails to explain any problem with GPs proposal. They would show you a transliteration of your name and ask you to confirm it. You would confirm it or not. It might match one or other of your IDs (in which case you would presumably say yes) or not (in which case you would presumably say no). What's the issue?

[Muromec]

You will compare the transliterated version I provided with the one you have already, it will not match and then what? Either you tell me I have invalid name or you just ignore it.

[lmm]

I think they were suggesting the opposite order - do an automatic transliteration and offer you the choice to approve or correct it.

But even if the user is entering both, warning them that the transliteration doesn't match and letting them continue if they want is something that pays for itself in support costs.