When you allow your customer to cash a check given to the customer by someone whose account is at another bank you are effecting a transaction initiated by the customer. Information disclosed in connection with "effecting, administering, or enforcing" a transaction initiated by the customer is exempt from the opt-out and disclosure provisions.
Honestly, however, I don't think this would even be deemed information "collection" in the pure sense. Either way, the privacy rules should not affect this procedure.