You are facing an interesting situation. However, without knowing the details of the contract, there is a potential detail that raises a red flag. If your organization owns the customer touchpoints, you must access, process, transmit, or store customer PII (Personally Identifiable Information), which the third party needs to provide customer self-service.
The question is, who is providing the customer PII data to the third party? If it is your organization, you must consider them as your third party and perform appropriate TPRM (Third-Party Risk Management) activities, including risk assessment, due diligence, periodic reassessment, and risk monitoring. You should also have a direct contract with the third party that legally obligates them to meet essential requirements such as cybersecurity protections and regulatory compliance, regardless of the third party's relationship with the bank. Suppose customer PII, provided by your organization, is integral to the service the third party provides. In that case, your organization is on the hook from a legal and regulatory standpoint.
If the customer PII is delivered directly through the bank and your organization is not involved in that process, you might have less to worry about. However, it is still strongly recommended that you carefully review all legal agreements with your legal counsel to understand your rights and obligations. I hope that helps, but I would love to hear other members' thoughts.