The additional information is helpful. It also would be helpful to know who filed the motion to stay the case and refer it to arbitration. By setting a deadline for your opponent to file an arbitration case, obviously the court was not merely making a meaningless deadline. It was willing to stay the case only for what it deemed a reasonable time. Courts pay attention to their dockets and will not acquiese to endless deadlines. It seems to me that therir are three avenues for you to pursue. They are not mutually exclusive and may be raised in one motion. They are: a motion to dismiss for want of prosection of the claim, a motion to dismiss for failure to adhere to an order of the court and possibly a motion to dismiss on the ground of laches. The latter almost always arises in equity cases. This case sounds like it would be more likely be on the law side. But with the merger of law and equity, I would still raise it.
Assuming that you succeed in the case being dismissed, then the question remains as to whether it is dismissed with prejudice or not. It is conceivable in instances where the court dismisses without stating one way or another that you might not know until the other side risks refiling. At that point, the court (perhaps a different judge) would decide. Whether with prjudice or not depends heavily on the grounds for the dismissal. Truthfully, I think it could go either way on any of the grounda suggested to support dismissal. A common belief in favor of arbitration is that it is less expensive and takes less time than court trials. In my view that is oversold and not necessarily true in more than a few cases. Nonetheless, the courts suggest that all the time in support of arbitration.. Therefore, the failure to pursue arbitration, even though the statute of limitations has not expired, could favor dismissal with prejudice on the notion that you have been denied the benefits of a speedy, less expensive arbitration. But, I don't think anyone is able to predict. Perhaps I'm too optimistic about your chances. You should see a lawyer in your jurisdiction to find out what the local lore is.