However, the API should also be able to support crypto payments, by getting a crypto address and then forwarding the request to e.g. a local instance of Metamask, instead of my bank.
Perhaps if the user has multiple bank accounts linked, they would have to choose one, to avoid leaking to my bank that I'm making a crypto payment.
Of course, a crypto transaction has a significantly lower chance of being compatible since there's no foreign currency team working at the bank - if the website wants a payment in BitWhatsits and my primary wallet is in DOGEMONEY 20000!!!!! then it's up to the user or at least the extension to figure out the routing. Probably, the website would offer a different button to pay with each supported cryptocurrency, so the user would choose. They'll be accustomed to that, being a cryptocurrency and all.
You put it better, just have any layer on top of IBANs. 0 need for crypto.