> The bank has to perform the authorization and identity checks, but the bank will not make them for you

We aren't talking about authorization, only about identity verification. I'm no domain expert but it is my understanding that banks provide these sorts of services. They certainly already have all the necessary information on hand both for practical reasons (security) as well as legal (KYC and AML laws).

> If it would, they will have to pay the bank for it ...

For the identity verification? Probably, depending on how you went about it. What's the issue? This is already a paid process we're talking about here.

For the additional legal assurance that I described? No, that doesn't cost extra. Please read what I wrote more carefully. It's a transitive property due to the penalties involved in addition to the degree to which the legal system and the bank care (at least assuming my understanding of that legal environment is correct).