At some point I said that I would accept the account number as authentication and they were unwilling to provide that. I don't know why they--a legitimate fraud-prevention department--expected to be able to cold-call a customer and receive identifying information. They should be actively working to prevent customers from turning over this information to "just anyone who rings them at home."
I ended the call and contacted my bank via their online banking system and via that system they vouched for the original call and provided a number to call back. I lodged my complaint that their own fraud department was calling customers without any ability to self-authenticate. Not sure what happened from there; I've not had to deal with that process again since.