It sounds as if perhaps the account that you are trying to access was not established in the name of the trust. It sounds like it is in the name of the decedent. If that is the situation then you would have to probate the Will that you have. It is probably a "pour-over" Will. That means that it leaves any assets that are in the decedent's name to the Trust. You may be able to use a probate procedure called Muniment of Title instead of obtaining Letters Testamentary. You need to consult with an experienced probate lawyer. You cannot do this yourself. It is unfortunately common that individuals establish trusts (often called Family Trusts or Revocable Trusts) and then fail to transfer all their assets into the trust.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.
Mr. Paxton's analysis is likely correct. If the certificate of trust would usually be enough, if the account was held in the name of the trust. Based on the documents the bank is requesting, the accounts were probably never transferred trust ownership. Even though you said this is a small estate, it is worth a consultation with a local probate attorney to get this resolved.
When responding to questions posted on Avvo, I provide a general purpose response based on California law as I am licensed in California. In reviewing my response, you are specifically advised that your use of, or reliance upon any response I provide is not advisable. I do not have all relevant background details or facts related to your issue / matter, thus I am not in a position to give you legal advice. Further, your review, use of, or reliance upon my response does not establish an attorney-client relationship between us nor does it qualify as a legal consultation for any purpose. For specific advice regarding your particular circumstances, you should consult and retain local counsel.