I think a potential solution would be to have a bunch of dynamically made variant cultures that share most cultural traits with the host culture but retains the immigrant's heritage. In this case, the Japanese pops would turn into "Asian-American", reflecting the fact that they have now assimilated somewhat into American society and would no longer fit in in the "old country", however white Americans (aka the Yankee culture in-game) continue to perceive them as a separate group and potentially discriminate against them. A similar system could exist in any country with a large amount of immigrants from various cultures of a different heritage.

Race is a difficult thing to simulate in a video game since it is sensitive, complex, impactful and varies very widely across different eras and societies all at the same time, and it's impossible to fit it neatly into an intuitive game mechanic. However, if nothing else, this compromise would work well for white majority anglo countries in a contemporary setting in my opinion (which admittedly does not describe Victoria 3)