I would argue that in this particular case, racism is a subsection of classism anyway. I've argued that an overt focus on race (identity politics) draws power away from a more egalitarian push towards socialism. You can find that argument here.
However, my dislike of identity politics aside, there is a simple argument for racism in the particular instance of monarchy.
Monarchy is a public institution, it is paid for by taxpayers - like all public institutions it should be held to account by discrimination laws. Given that positions in a hereditary monarchy are hereditary by nature - the position of 'King' or 'Queen' can only be held by someone mixed race. That is as good as it gets. The structure of monarchy, unlike that of all other government institutions is racist by design in that it implicitly excludes people of a certain skin colour by the rules that codify it.
Separate arguments can be made about only having white men from Eton as the Prime Minister - such things aren't baked into the rules themselves, they're just a British tendency to elect stupid posh people out of bizarre social class obligation. That may suggest the general public is both classist (and racist) but that is neither here nor there.
There are no stats or sources quoted because I'm not an academic writer and that's not my style. It's an attack I get a lot, but I prefer being a provocateur and in this case I don't think stats or sources are relevant. The argument stands on its own merit - the main critique I've got is... does this make other monarchies from around the world racist? Yes, it does. Any hereditary monarch that passes unquestioned political power through genetic succession from the Norweigens to the Japanese emperors (and even the Kim family of North Korea) selects against people based on race.
That is racism.
Which is why I'm so confused when so many woke people stand outside waving flags for it and getting so delighted when Harry breaks wind or Wills says something positive about helicopters. Baffling.