Do it P2P find a person you know which selling his Bitcoin then buy for him directly .in that case you will never need to send any identification from exchange the only person that will know you is the person where you buy your Bitcoin.
You should read the main post once again. He doesn't want to expose his identity, it doesn't mean that he only don't want to submit his KYC documents, but also he doesn't want to expose himself as a Bitcoin user to the third party.
It's good move that OP is looking to buy Bitcoin anonymously. I believe OP already learnt how you should buy Bitcoin anonymously. Just an additional suggest, please choose a right wallet to keep your fund secure. Better to select a hardware wallet or a noncustodial wallet. You may take a look on
this thread to realize which wallet you should choose.