I think the risk of that approach is that if you attempt to copy their accent too literally it can sound like mockery, especially if you are clumsy in your imitation. Like you’re breaking out of your own accent on purpose because you think their name spoken in their accent sounds silly, and by repeating it in an exaggerated way you’re demonstrating how silly it sounds to you, and that kind of response can be interpreted as mocking or sarcastic.

I think it’s safer if you try to strike at most a middle-ground between your own accent and their pronunciation, use it as guidance for the sounds but still keep it clearly in your own voice. When somebody has an accent I expect my name to be spoken at least to some degree in that same accent, so it’s not going to need to be an exact facsimile of the sounds I made.

That’s my thoughts anyway, as a native English speaker.