you can almost certainly do a chargeback on your credit card, then they have to come after you, and their legal costs will hugely outweigh any payment they are seeking
Check your credit card terms of service carefully. Cards do have consumer protection, but generally it isn't about this type of situation. If you owe the fee then it is fraud to do a chargeback and you can get into legal trouble (though odds are as the other poster said: they will just send you to collections and it will hit your credit report).
I agree that this type of change to contract terms is not acceptable, but unfortunately you need a lawyer to fight it.
