If someone were to crossbreed two differently-colored vivillon, would the offspring simply look like one parent or the other, or do the wing colors have incomplete dominance and the child would end up with a mix?

Actually, wing color depends entirely on environment, rather than solid genetics. More specifically, the temperature and general weather that the egg is exposed to upon being laid will dictate what type of vivillon wing pattern the hatchling will have later in life. So for example, even if you had a tundra vivillon mating with a sun vivillon, if the environment that they’ve nested in is temperate and home to an abundance of trees native to Japan (ideal conditions for elegant or meadow vivillon), then the resulting hatchling will end up being either elegant or meadow in variation, rather than either tundra or snow. Scientists believe this is a sort of evolutionary survival technique in which the vivillon have essentially managed to adapt traits that enable its offspring to survive, even when displaced or born during the vivillon’s life-long migrations.