You miss the point or ignore it completely. If the man's name is Patel pronounced "Pah-tell" that how you pronounce it, even if everyone you know says, "Pay-toll." You don't need to change your accent, just say the guy's name correctly or best you can, it's not that hard. I know many people with difficult names from India, Pakistan, Africa, etc, but it would be an insult to pronounce their name incorrectly just because it's said different in the U.S. (e.g., "JOE-hand-son" vs "YO-Hahn-son"). If you're unsure or have trouble, ask them for a phonetic spelling until you get it right.
Native (and nearly native) speakers can easily sound out a word (I can't roll my "Rs"), but why should we expect tehm to change their accent to speak a fluent language? I'm sure many would get upset if Spanish speakers told us we shouldn't speak English words with an English/American accent in a Spanish country.
Morover, in this case, the woman SPEAKS Spanish fluently, as well as someone who only spoke Spanish. She's not changing her accent, she's speaking Spanish.