You can fill out the Account Verification Form here:
https://us.battle.net/support/en/art...ification-form
They require a government issued photo ID to proceed with a name change. I have never had any luck going to a GM or calling their customer service to change the name on an account, I have always had to use this form. As long as its obvious that the current name on the account is not your real name, like an avatar/user name, they should change it.
Adding that you want to change the name for security reasons if you were ever to get hacked or have your account compromised wont hurt either.
This is not true, they would not close the account. Please don't spread rumors.