I am not sure if the paid version allows you access them anonymously.
Even if you can, how can paying with your credit card keep you anonymous?
Well, you're right.
That doesn't make sense at all since you're still required to provide all of the details of the CC company that you're going to use for paying.
How about exploring with other AI websites that have the same feature as chatgpt and probably they have this feature, I don't know. Do the searching, friend.