Maybe send a letter officially

telling them that you challenge the validity of the charge and refuse to pay it. provide the documentation again.
From this point forward, ask every name that you talk to and record the conversations if you can. Do as much as you can through writing so that you have a permanent record of it.
I would not pay it – it is a matter of principle.
What is the worst thing that could happen if you ignore it? Do you think that they’d sue? If they do, you’d obviously win because you have the documentation that they are wrong.