Genetics simply aren't the same for primitive strains of the horse family. I can guarantee you he wouldn't test positive for palomino, or at least, not in the way we see it in horses. This is the reason zorses and mules and zedonks are hybrids (also ligers). When crossing into seperate families, the rules change, and nothing "applies" anymore. You're mixing two completely different sets of genetics.
Remember that dun in our horses as just a branch off of the primitive coloring you see on zebras and true wild horses - the dorsel stripe and zebra striping isn't just there for fun, it's because at one point this genetic expression of color survived down the lines into our modern day horses, showing how dominant it can be as a genetic expression. As you can see in the wild mixes, some shading will come through and look like bay or palomino, but that dun expression is so strong coming from a horse that only knows how to be striped that that's what you'll always get!
I do find the pintos ADORABLE though: