This is a curious case and I would appreciate hearing more detail, as in, what sort of bank? We've had some discussion around here recently about USAA. Anyone, it seems, can join and get an account /password. If you log in, you can go to Hertz, and there's a Hertz welcome page for USAA members. You can also just plug in the USAA CDP and use it as well. USAA has stricter requirements for accessing their insurance products but they freely dispense with the CDP to anyone who has an account. But I wonder how it would resolve it they challenged me? Can I prove my membership by logging in in their lot? Probably should print out my credentials, right?
I often use AAA, and I'm a AAA member with a current card, so that should not be an issue. I also often use the CDP for an organization for government employees and retirees (state and federal) AND the same for employees / retirees of non-profits (for which I qualify, and I have a company ID). But when I joined that organization (which may be defunct now, though the CDP still works) I wasn't challenged in any way to provide credentials.
I'm just curious what level of proof we need to carry if they are going to start to challenge CDP use, which is of course entirely within their rights. Interestingly, in 35 years of renting from Hertz, I've never been challenged.