Lucy is required to be the third member in your fighting party, you'll acquire her by defeating the sub boss BoJonn for the second time in the arcade and then following him through the hole in the wall. Lucy will be just around the first corner on your path.
The three costumes needed to be worn are "Robot, Knight and Statue of Liberty" and it's not required for any specific character to be wearing any particular costume.
Winning the competition will also give you a costume piece towards creating the Unicorn costume which is required for the "Master of Disguise" achievement.