BTC ain’t that anonymous as it use to be anymore. Meanwhile you gotta sign up with real data if you want to buy bitcoins. Best way would be to create a wallet and buy from people who already got bitcoins. Creating a wallet do not afford to publish real personal data, buying bitcoins from official resellers mostly do. Purchasing from people who don’t care who you are won’t require sharing real personal data.
Or, how about this:
1) You buy BTC through a source using real data (such as CoinBase)
2) You transfer it to your wallet
3) You make purchases from the wallet