I mean, the judge is going to respond by sanctioning the attorney(s) for frivolous litigation. If your legal argument (for why the case shouldn't go to arbitration) has been rejected in court, initiating even one more case before the same court with the exact same argument could justifiably be considered a waste of the court's time. Initiate several of them and you're going to be fined; 10,000 and you're probably looking at things like contempt-of-court charges and disbarment. That's aside from the fee to initiate each case, as Chaebixi mentioned.

That said, if you're willing to risk all that to make a political statement - i.e. to pressure legislatures to change the law around arbitration - I can't rule out that it could be effective. But I suspect that other avenues for making a statement would be cheaper and more effective (and avoid the stigma of abusing the court system).

For example, how about just taking out TV ads? Find some maximally sympathetic victims of the arbitration system, and have them tell their stories. Less dramatic, but right now I think there isn't all that much public awareness of the issue - the New York Times' series notwithstanding. Arbitration is such a classic "big corp screwing over the little guy" story that I think a large number of even Republican-leaning voters would oppose it, especially the Trump camp (ew, but still), if they heard the full story. But most of them have probably never heard of it, or have only heard one-sided accounts that emphasize how class-action suits cost businesses and don't usually win plaintiffs all that much money. (Both of which are true, but ignore the fact that class actions are effective at holding defendants accountable for violating the law, and getting them to change their behavior - whereas a trickle of arbitration cases can just be dismissed as a cost of business.)