I work for a bank that operates on the coast.
There is NO law in NC prohibiting open or concealed carry in banks. The law is the same as it is for any other business - they must use a conspicuously posted sign OR management must ask you to leave if the site is not posted. I actually have gone over this with our physical security team since some staff wouldn't have an issue and others would freak out and hit the panic button (likely opening us up to a lawsuit for harassment).
Now - even without knowing what bank it is (it might be the one I work for after all), I would go in and close every account you have with them and tell them why. Or ask the branch manager who their boss is and contact them. They are often known as Regional Operations Officers and are responsible for many stores. Personally, I would comply (since by law you have to) but either stop doing business with them or talk to the right people and get written confirmation regarding whether carry is allowed or not. I can tell you, my organization would NOT allow a branch manager to determine his/her branch was "gun free". While the law does, if you go above them you can perhaps get the policy changed for that manager from "on high".
If you don't mind - would you share the name of the bank? If it is mine I can help by steering you to the appropriate people.