I'm in no position to talk about Japan's ethnic diversity or lack thereof, but there is a possible way to get at least more diversity on certain levels. When you look at the linguistic map of Korea, all the dialects and such, one big thing you notice is Jeju. Jeju's spoken language is officially recognized as a dialect of Korean, but in many ways it's unintelligible to people on the mainland. There are also other differences in vocabulary and the like between the historical 8 provinces, which indicates how isolated they were, mountain boundaries and the like being natural ways of keeping even the many Korean populations relatively isolated from each other. Perhaps if you accentuated that, like had some sort of scenario where people mingled and crossed between provinces even less than already, that would be a start.
Foreign invasions on the other hand, would need to be foreign occupations, and for longer periods of time. One possibility is if say, the Mongols and the Yuan Dynasty, instead of just making Korea a vassal state, actually attempted to annex it for a hundred years outright, and encouraged settlers to move there and such. On the other hand, that would run into problems once the Yuan or Mongol control falls, because then I could easily see people who are still ethnically 100 percent Korean going all ethnic cleansing, if not outright genocide against people who are either bi-racial or who intermarried with any occupying force. That's one of the big problems for making Korea more diverse. Hell, you'll even see it today with North Koreans. NK women who come back from China pregnant better hope to God that they can keep that a secret somehow, along with the baby's mixed heritage, otherwise the baby's dead and the woman's prob off to a concentration camp.
But running off of that thread, just maybe if you had a situation where a unified Silla controlled large parts of Manchuria, Goryo loses parts of it, Joseon gets some back, etc. You would need a border situation that fluctuates a lot more often for basically all of the last millennium.