This is exactly what small claims court is for.
Small claims court is exempt from arbitration requirements (which are primarily aimed at avoiding class action suits). It doesn't require you to hire a lawyer, and probably won't get your account automatically nuked the way a credit-card chargeback would.
I don't think you even need to go that far. Just refute the charges with your credit card. Very high likelihood of a successful refund since they already acknowledged their error in writing.
If you file pro se and even if you've agreed to ten thousand arbitration clauses, they'll at least have to spend $200 on a lawyer to respond.
So, you can waste as much of their money as they wasted of yours.
You're totally right! Please refer to paragraph 213 of your service agreement, in which you agree to binding arbitration with an arbiter of our choosing at your cost. I hope this answers all of your questions! Have a wonderful day!