As a regulated financial institution, you are required to have a complete inventory of all third parties, including other financial institutions. Even though you may receive limited responses to your requests for due diligence, you will need documentary evidence that you tried.
You may also want to include questions about the scope and maturity of their third party risk management program in your due diligence process. They will likely respond to these questions providing they are not too invasive.
If you are satisfied that their program is robust and compliant, you may wish to use this as documentary evidence, along with a "reliance" rationale that the bank is also required to comply and subject to regulatory exams, as rationale for determining that they are low risk. This can be signed escalated and off by Ops Risk/ ERM.