The character wasn't white, he was hapa. And this was a trait added to the character only after they decided to cast a white actor instead of an Asian one (having passed over, most prominently, Bruce Lee for the role).
So they had a character that was half-Chinese but still cast a white actor (and the character was only half-Chinese because it is easier to whitewash half an Asian rather than an entire one).
There's a reason that they tried to set things right a little bit in the two Kung Fu movies that have previously been made by having Brandon Lee (Bruce Lee's son) play Caine's son. In the first (Kung Fu: The Movie) he plays Caine's unknown son and in the second (Kung Fu: The Next Generation) he plays Caine's grandson.
So yes, the mixed-race nature of Caine is probably ingrained in the story line (though probably for racist reasons) to the point that it won't be removed. But they can still get a Chinese, or at least hapa, actor to play the role.
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