There aren’t really any widespread hard feelings against Japan for the war. OTOH, a German princess (or pseudo-princess) might - despite the jokes about the royal family being German - be a problem.
Actually, a non-British princess would be a problem regardless of the nationality, but, if they were sensible and had a long relationship like William and Kate, and it was just as obvious that it was a real loving relationship, the public would adapt.
That only goes for William and, to a lesser extent, Harry. Nobody beyond royal fanatics (and the royals themselves) cares who the rest of them marry.