Just brainstorming:
Perhaps you each team you pick could have two factors that combine to give you their monthly score:
1. Average percent of perfection (just as you described)
2. Handicapping factor. (needs better name). The higher the percentage of people who pick that team, the lower this variable goes. Teams with fewer people picking them would get a higher multiplier. There would need to be a limited range for this so you couldn't simply pick a low-scoring team that no one else picked and win. IMO, this should be an unknown variable to the players until announced after the cutoff time.
There would need to be a lot of work put into the formula to make it simple and fair.
IMO, predicting the percentage perfection score seems to add a lot of complexity on the player side. It would be easier to just pick 5 teams and more people would want to play.