Sorry for missing your post, but hopefully I can still be of help a year and 2 months later;

Once the switch has been activated, the BIG SHOT meter will appear in the battle area. For every attack you survive while still standing firmly inside the minecarts, the meter will increment until it is full. (You do have to be standing inside the minecarts. You can't be on the edges, on the front, you have to be properly standing in them. Any of the three will do.)