How do I prove to a bank that I'm someone's attorney?

You show them the registered LPA. The bank will need to see the original or a certified copy of the LPA showing the OPG's registration stamp on every page. Banks have their own processes for adding attorneys to accounts — usually a visit to a branch with the registered LPA and your own ID, and sometimes a separate "Power of Attorney registration" form the bank requires you to fill in. The Big Four UK banks each have slightly different processes, and most have a dedicated Power of Attorney team you can call to set things up. The whole thing usually takes a few days once you have the registered document in hand.