This is not a national bank.
To give a small update: My co-worker tried to call the customer back yesterday. No answer, and no returned phone call. Our manager said she has to try again today and again on Monday. 
On a more positive note, after this debacle, the bank IT team took ten minutes out of their busy day monitoring internet usage for the approximately 70 employees that we have and changed the default setting so that the option to print the balance isn't available any more. They hung signs instructing that anyone wanting a balance must specifically request one and show i.d.
I would SO throw management under the bus, if i were that coworker.
I'd be saying, "I'm so very sorry this happened--I've been dreading it for years, because the computer system was set up to default like this. My coworkers and I have known for years that sooner or later, one of us was going to get tagged by it, and the balance would slip through when the customer didn't want it. We've been arguing for months to get them to change it, because we knew that customer was going to be affected. We couldn't get them to act.
"I'm very sorry that you were the customer affected, and I do apologize. if there's any silver lining, it's that your experience has *finally* gotten them to pay attention to us. And I wanted you to know that the IT team has changed the software--it took them only 10 minutes, which is far less time than you've spent in bringing the problem to their attention--and WAY less time than we've spent arguing for the change.
"I really am sorry that you were affected by this, but I hope it will help to know that they have definitely fixed the problem and it won't be happening to anyone else."
I'd be apologizing for management.