We can't answer questions about the different bank and prepaid cards Turbotax uses. Your can try and manage the phone tree to get a human.
In general, if you are calling the number on the back of the card, then you are calling them and it is safe to provide personal information to verify your account. And I expect that you need to provide some personal information to prove you are you and not someone who stole the card or found it on the ground.
Maybe if you can get transferred to a human they will let you provide just the last 4 digits or verify your identity in some other way.