Seems like the point of any petition on the matter would be to say, in effect, "Here's what I would do if I were you."
If the point of the poll is to ask that question, then I would take it to trial. I don't say that because I want to see AP get taken to the cleaners specifically, but more to the point I think it would be useful to use this as a warning to the rest of the fan film community. It's time for C/P to fire the shot across the bow to everyone - play nice with us and you can do what you like, but as South Park's Cartman would say, "respect our authoritay".
In my ideal world, C/P would win a large monetary victory, would then spend a couple years extracting as much financial pain on AP as possible to get the money once the appeals are done, and then quietly drop it (once I've at least gotten back my legal expenses plus some), since actually getting all of it over a film that was never made is secondary.
So that's my opinion, but at the end of the day - Who am I to be inserting my opinion on the matter - even indirectly via petition - into this conversation on C/P, and why should they care what I think? Long term, no one is going to stop watching Star Trek and buying merchandise no matter what happens to Axanar. It will blow over, and won't affect C/P's bottom line on the franchise.
I think the whole affair is kinda sad, because I fear the outcome of all this is going to be that C/P makes a VERY restrictive set of guidelines (to prevent exploitation of loopholes) that in the end kills off fan productions almost entirely.