For background characters with only a couple of lines, I don't mind using coloquial language.
"Y'all come 'round back. Gots som'in to show ya." I don't care how they make the character sound since it's a one off. The accent is whatever my MC hears. For MC's, I'll use a light smatterin' of accent, words like "Y'all" or whatever would suit the character, however I use them sparingly because they can throw the reader out of flow. I'll write, "Y'all come around back. Got something to show you," he said in a southern drawl so thick flies could stick to it like molasses. Let the reader insert their own accent. As for being racist, unless one is making fun of the accent, using it as part of a character ain't racist. This is coming from a Georgia boy living up in New Yawk (New York), surrounded by Yankees (anyone from above the Mason-Dixon line). Trust me, my accent attracts more laughs of derision than sighs of sexuality up here. I've even had a manager call me both stupid and lazy based solely on my accent (though thankfully not for long. It's amazing how quickly things turned around after I was named head of compliance review. I still see him sometimes, though I never answer yes when he asks if I'd like fries with my order).
You obviously don't know what you're talking about.
Twain was reproducing the African-American dialect he heard. Yes, it existed. Read Charles Chestnutt, an African-American author, and any of the African-American authors of the 19th-early 20th century. They did it too. There are African-American speech patterns of today. I had a black boss who used the pronunciation "ax" for "ask." If I wrote about him in a book, I'd use that pronunciation.
Cringe away.
That they did, and in some parts of the country, still do. And it wasn't just African-Americans that did it during the time period. The accent and methods of speech were also found among the poor white populations, indicative of a wide spread, pervasive lack of education without respect to race. You want to hear the accent for yourself? Go visit the Appalachians. People really should study history more instead of jumping on the revisionist bandwagon.
As for the OP. Making fun of accents isn't limited to Asians, but if you look at American History, Asians have always been portrayed in a negative light, much more so than any immigrant group. We have created laws specifically aimed at the Asian American populations in the US, in specific violation of the 14th amendment. We didn't even try to pretend Separate but Equal with them and even went so far as to enprison them during the war without regard to their civil rights.
It's a world wide phenomena, an us vs them mentality. There's an ass in my office who thinks it's hilarious and entirely appropriate to make fun of my southern drawl, and never considers what I may think of his accent.
Personally, I say the OP should write it that way. Don't go out of your way to offend people unless you let them know ahead of time it's part of your plan, and if you personally find it offensive then by all means, remove it. If not, and a listener is offended by it, oh well.