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Seeing as Buffy can be a nickname for Elizabeth, I don’t see why Kai can’t work as nickname for Katherine. It’s a stretch, but I think you can make it work.

It can. But it wouldn’t really sound right, and I wouldn’t prefer it. Pretty different from each other.

Too far.

No

It works.

If it's pronounced like the name of the letter K, which would be Kay, then fine. But if it sounds like Kai, and rhymes with high and sky, then it is too much of a stretch.