Should you copy a person's accent when pronouncing their name?

https://lemmynsfw.com/post/30973507

Should you copy a person's accent when pronouncing their name? - Lemmy NSFW

For example, an English person might introduce themselves as “Bob”, whereas an American with the same name might introduce themselves as “Bahb”. (Sorry, don’t know the phonetic alphabet but hopefully you get my gist) Should you pronounce those two people’s names the same, with your own natural accent, or should you copy how the person says their own name?

I think it depends on whether the sound difference actually makes a phonemic difference in the source language—like, would a native speaker hear it as an oddly-pronounced version of the correct sound, or as a distinct, different sound in that language?