Banks could sell single use/secured throw away cards that you buy like a money order. You purchase the amount of funding and it is written to the card at the bank. Your pin would be the security that unlocks the card and the card wouldn't have to have your ssn or credit info on it. Once the funds are used the card could be discarded (no pun intended) or renewed with more funding. Think Walmart gift card.
Now, when my antivirus subscription expires or I want to purchase that game or some ink I go to the bank and buy a purchase card and next time I'm online I type the number and unlock code into the website and throw the card away.
Since the card would be coded at the time of purchase the website knows it is good and the money is transfered from the bank to the sites bank same as normal. The billing/shipping info is what would be used to track your order. The bank computers would only track the SOLD card numbers.
The entire card purchasing operation could be automated for convience in the ATM system.