Something to bear in mind is the fact that the attitudes weren't as, um, clear cut (the other metaphor I was thinking of really doesn't suit) as you might expect - simply because prejudice was institutionalised doesn't mean everyone subscribed to it. Sure, probably most people did, especially upper-class types. But overt racial hostility was, to my mind, very rare. You could even be great friends with people of colour, if my reading of The Spirit archives is accurate - even if black people are little more than offensive caricatures in these ancient comics, they're fighting for the side of good (well, one is, and I don't recall any black villains from my reading), thus being better people than the gangsters the Spirit is perpetually foiling.
So it shouldn't be too hard to edit out the racial tensions altogether, especially if a gang of PCs tends to do PC-y things like drive their own cars, don't spend long hours roleplaying breakfast at a restaurant, etc etc. And when tensions do crop up, the best people to wield them are, as you've said, villainous Nazis.
That's if you want to do without them (and I'm pretty sure I would - makes me feel queasy). If you do want to keep the sordid details of history, my statement stands.