Carrot losing to Perospero is important, because she needed to learn that revenge is stupid. Pedro wanted Carrot to help bring about the Dawn, and she's trying to get revenge and murder a guy who didn't even kill Pedro. Perospero NEVER tried to kill Pedro, and Carrot is still obsessed with blaming Perosphero for Pedro's death. Up until now, Carrot has COMPLETELY MISSED the point of Pedro's sacrifice.
Carrot needs to get over her petty revenge and focus on doing what she really wants to do: fulfill Pedro's Will, and help the Straw Hats. Carrot needed to lose to Perospero, to realize she was going down a dumb, dark path. Now, I think she's about ready to join.