I don't usually get involved with these kind of threads, since I don't post here a lot, but I've been musing over this all day ...
I'm Italian-American. A significant portion of my relatives are immigrants or still live in Italy. If the OP had related the same story, except she was in an Italian restaurant, and had related the broken English used by an Italian immigrant, I would not have been at all offended, unless it was done maliciously or mockingly. I didn't pick that up at all here.
I love my relatives. They almost all speak broken English, to varying degrees. When I relate stories about them, or write about them, I try to capture their language as it is spoken; I don't edit it for grammatical correctness or proper pronunciation. That's not who they are. Their usage of the language, however improper or "stereotypical" is part of what makes them unique and beautiful. To correct their language for them doesn't just flatten their character and personality, it's actually a bit insulting.
If my Zia (aunt) says to me, "Oh, dio, che bella bambina! She want a sangwich?" and I want to write that down in my journal or as part of a story, I'm not going to write, "My, what a lovely baby. May I make her a sandwich?" That makes her sound like a completely different person, even changes her personality. It would leach her ethnicity, which is a big part of her, completely away.
I don't know; I would like to think that if the shoe were on the other foot, and my relatives found my broken Italian charming, they would feel free to share that as part of MY personality and not feel like they had to pretend I'm not an American because it would be stereotyping.
Now, if someone giggled at my father, who looks Italian but has perfect English diction, and related his language to reflect a thick immigrant-speak, THAT would be stereotyping. If someone looked at him and said, "I bet he's in the mob and loves capicolla," THAT would be stereotyping. If someone related my aunt's speech, not as she actually speaks it, but made it up to sound more "Italian," THAT would be stereotyping.
I guess what I'm saying is that it's not stereotyping to portray someone as they are. Would we be having this discussion if the OP were describing a Texan twang, or a Boston accent, or Long Island talk?