A Letter of Authority refers to the document (or Letter) that grants the personal representative authority to act on behalf of the estate of the person that died. Probate refers to the court procedure by which a decedent's estate gets administered after death.
Therefore it is not known that what exactly is your question, because it is not known that how your mother's name was included in the letter of authority.
Are you by anyway referring to the list of legal heirs on the petitioner's side as letter of authority?
Your mother is not a legal heir to your grandmother, hence she cannot be included as a petitioner seeking probate of your grandmother's Will.
The bank lawyer is right in mentioning that your mother not being a legal heir she cannot be added as a party to this probate especially during the lifetime of your father.
In any case the probate has been granted by court hence the probate of Will should not be a problem.
You can approach a different bank for loan.