Listen up, because I'm about to fix all of the race problems The Mikado may or may not have. My solution is simple and elegant. Set The Mikado in Scotland.
When Sidney Grundy wrote the libretto to Haddon Hall for Sullivan to set, he used a Scottish character to justify some extremely strained rhymes. It is, therefore, I think excusable to make the first line of the show "If you want to know who we are, we are gentlemen of Scotlan'." And all occurrences of the word "Japan" are replaced with "Scotland" and ditto with their respective adjectives. In the case where a three syllable substitute for "Japanese" is needed, the word "highlander" will do, and I don't think anyone will complain.
As for names, well all we need to do is stick on a prefix. Pooh-bah? MacPooh-bah. Nanki-poo? MacNanki-poo. Pitti-sing? Pitti O'Sing. And if the accent is thick enough. No one will care. They'll just hear a sound they assume is a name. And, of course, we have Mr. MacAdo.
And the best part of this is that the pentatonic scale, which Sullivan uses to establish a Japanese style for the music, is also used in a lot of Scottish folk music. Miya Sama will sound thoroughly Scottish coming out of a bagpipe!
Doesn't it really all seem so obvious now?