The only guaranteed way to stay kyc free is to never transact with an entity that forces kyc. Mixers only obfuscate the transaction path and there's no guarantee that this obfuscation won't be unscrambled in the future.

The thing is, you have to expose your private key to the new chain. Only if you swipe your key(s) you'll be able to claim those coins. So you basically have to swipe, move all your BTC coins and then you can do with your forked coins whatever you want. As for doxxing your non-kyc... you just dump them on a non-kyc exchange.