In the case of a living trust account, the “customer” is the trust. Further, if the trustee is a third-party (in this scenario, a law firm), the CIP rule also provides that the bank may need to obtain information about individuals with authority or control over such an account in order to establish the true identity of the customer. We obtained and recorded a fully executed copy of the Trust Agreement to verify the trust. We also performed documentary verification on the attorney designated as the authorized signor on the account. Is it prudent to also obtain business documentation on the law firm ( e.g., Partnership Agreement) as well as identify all beneficial owners of the firm?

There is prudent and there is more prudent. Only your institution can decide the line between prudent and overkill. That decision might be made (or changed) if there's significant push-back from the prospective depositor. I am surprised that push-back didn't come when you asked for CIP info from the attorney (just because attorneys like to push back as a general observation). And I know there are some bankers who would be aghast at your insistence on receiving and copying the full TA. In some states, the law says an affidavit as to the existence and particulars of the trust must be accepted, and some banks argue that having anything more puts them at risk of knowing more than they want to and perhaps puts them on notice if a trustee does something not permitted in the TA.

Generally, only individuals and banks with trust powers can serve as trustees. I would have been more concerned about accepting an account with a law firm as trustee than anything else...