Sure you can offer to fix it, but since you don't know squat about the system (save for a small flaw at the surface) and they have teams of developers who do, they won't be interested in paying you to fix it. Offering to explain the bug for a fee won't be blackmail unless you threaten to reveal the bug to others if they don't pay up.
Be a decent chap. Send 'em a nice letter explaining the problem. It's your bank, remember, and they're humans like you; work with your service providers to improve the service. Assume you're not the only one who knows about the problem, that someone who also knows isn't as nice as you, and it's YOUR bank balance that is at risk.