This was talked about in one of the threads earlier. If someone were to do it, I expect it would be rebuttal-type briefs towards the other briefs.
Doubtful the other sports leagues file and get tainted with this mess. But in theory they could, to argue that they negotiated away article 46 powers in exchange for something else, and that they believed those powers were worth what CA2 ruled them to be, and that therefore reversing the decision would reset the bar. This would counter Olson himself and the Patriots.
They could find a corporate lobbying group to rebut the AFL-CIO brief.
If there are any scientists who want to get their reputations involved and want to rebut the scientist brief and support Exponent, they could. Possible Dr. Marlowe could file to defend his work as peer-reviewer, but he may be considered a direct party so may not be allowed.
They could find another arbitrator who says all those far reaching conclusions that Feinberg claims will happen, won't actually happen. Hey, Roger Goodell is an experienced arbitrator, he could do that!.... Seriously though, I think this one would be most likely of all, as some short-sighted arbitrator would be happy to have his powers increased (without realizing his case load will fall to zero as people start avoiding arbitration altogether).
They could also try to find other law professors who think there aren't any implications to counter that brief.
As mentioned, I believe the arbitrator would be the likeliest to happen, as 1) that brief is the most damning, and 2) it's the one that might be the most contentious within its peer group.