Is there even an exchange that doesn't implement KYC? I highly doubt it. You'll have much better luck buying Monero from a third party and using that to buy back BTC, which you will sell to another third party for fiat. Exchanges are a no-no if you absolutely require privacy. Either way, there are too many steps that it may be more ideal to use mixers like Chipmixer instead.

Also, I honestly don't think the government will single you out unless a) you're dealing with an ungodly amount of money, or b) you're into shady stuff. I know that exchanges are hard to trust nowadays with that entire IRS mess, but it also proves the government will have to go through a court to obtain your data. It's not that simple.