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.