The only thing you can do to stay anonymous is using decentralized exchanges so you will avoid to send your ID to third party. My suggestion is you search for your local seller who sell Bitcoin and receive fiat from your bank.
You don't have to tell them who you are and your bank only know you do usual transaction. We can not avoid verification process if we use centralized exchanges but we have many options to buy Bitcoin. With the presence of decentralized exchanges, that help us to buy and sell Bitcoin without verification.
Local sellers are usually cooperated with centralized exchanges.
They might not ask KYC to trade with them, but if the banks and centralized exchanges know the sellers, they might watch the buyers too. Because it's traceable the banks know that Bitcoin seller send fiat money to your banks, so they likely know you have Bitcoin.
Most people avoid to submit KYC because they don't want to suffer data breach, so they won't have any problem in this situation. It's different if someone want to be completely anonymous, because this leave a trace.
Just do it with cash and nobody is going to be able to "watch" you. Doesn't matter where the other party sourced the coins.