>But you can compare the transliterated version to make sure it matches

No you can't.

Add: Okay, you need to know why. I'm right here a living breathing person with a government id that has the same name scribed in two scripts side by side.

There is an algorithm (blessed by the same government that issued said it) which defines how to transliterate names from one to another, published on the parliament web site and implement in all the places that are involved in the id issuing business.

The algorithm will however not produce the outcome you will see on my id, because me, living breathing person who has a name asked nicely to spell it the way I like. The next time I visit the id issuing place, I could forget to ask nicely and then I will have two valid ids (no, the old one will not be marked as void!) with three names that don't exactly match. It's all perfectly fine, because name as a legal concept is defined in the character set you probably can't read anyway.

Please, don't try be smart with names.

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?