I apologize as my initial response may not have addressed all of the issues that the bank raises.
The only way to add or remove someone as a check signer is for someone who is an existing check signer to fill out a new signature card. If you and I were partners in the LLC and you were the authorized signer, the only way to add me as a signer would be for you to complete a new authorized signer form that adds me to the account. Just because I’m a member of the LLC doesn’t mean I have an automatic right to sign checks.
In terms of limiting the amount of a check that someone can sign, you should be able to accomplish that by requiring the use of two signatures on a check for amounts in excess of $XXX. You can also have that printed on your checks. The drawback is it will limit your ability to sign checks. The instructions on the account may say that for checks in excess of $10,000 two signatures required. In cases like this, you will need more than two authorized signers on the account, e.g., one of the check signers is hospitalized or on vacation, you still may need the ability to issue checks in excess of the limits you set up.
Their comment about an authorized representative on the account being a de facto member of the LLC may be referring to a legal term called apparent authority. This is when a principal takes some action with respect to an agent, employee, etc. that may lead a third party to believe that the agent, etc. has authority to bind the principal. For example, you sell products to retailers and use a manufacturer’s rep for the selling process. You give the rep business cards from your company that lists their name and a title of VP of Sales. To a third party, it would appear that your rep is an officer of your company and able to bind your company to agreements with third parties. If the rep created a card on their own without your knowledge and pretended to be an officer of your company, it would not necessarily bind you to an agreement since you did not initiate the action that caused the confusion.
As to this type of situation being created because someone is an authorized signer on your bank account, I would tend to think not. BUT, I am not a lawyer and would yield to the attorneys on the forum to comment.
Feel free to email me if you have any questions that you want to take “offline.”