What documents does a bank need to transfer a deceased person's account to a beneficiary?

I am the beneficiary on my mother's bank accounts. 2 of the banks accepted the death certificate and transferred the account ownership to me, but a third bank is asking for a short certificate AND a letter from a lawyer stating that I am the rightful recipient. I believe they are being overly zealous in protecting themselves. The whole reason my mother named me as the beneficiary on her bank accounts is so I would avoid having to probate the will, which I have, but see no reason to use. There is no property other than the 3 bank accounts. The amount in question is under 40K.