I think one of the important takeaways here is that there often is absolutely no need for racism to make many funny bits work.
My case in point would be comedian Bill Dana, who used to do the character of José Jiménez on the Steve Allen show in the 50s/60s. After a few years of doing this character (and after public protests by Hispanics), Mr. Dana appeared to have grown a conscience, and later made a public apology at a cultural pride festival.
The thing is, the bits work great without the racism! There was never any need for it.
You simply have a character playing the fool - and I don't even mean mentally challenged - and it works.
Ed Sullivan: "Well, now I see you have some of your space equipment with you. Uh, what is that called, the crash helmet?"
José Jiménez (as an astronaut): "Oh, I hope not."
Ed Sullivan: "Now, what do you consider the most important thing in rocket travel?"
José Jiménez: "To me the most important thing in the rocket travel is the blast-off."
Ed Sullivan: "The blast-off."
José Jiménez: "I always take a blast before I take off. Otherwise I wouldn't go near that thing."