That wouldn't matter anyway, as the start buttons work in any of the 3 modes.
Couple things:
1) Is this P2 Start button ONLY wired for P2 Start, or are you using it as a jump/fire button, too? Sometimes one signal can override another, and cause a button not to work properly.
2) If you are using quick-connects on the button, try swapping the ground wire with the one on the working P1 Start buttons to see if that affects anything.
3) Swap the entire switch out with another "just because". You may test it with your meter and it goes down to say .040, but that might be just off .000 enough to keep the board from recognizing it as a low.
4) Do a continuity check from the "hot" wire for the switch all the way back to the pad on the board for that signal. I had a wire once that broken inside the insulation and wasn't connecting.
5) When you put it into Input Test, do you get any signs of life when you push the button? Any intermittent working or anything?