In conclusion, in order to unlock all of the costumes, you need to complete both Remi and Choux's Story Modes, and every sub game mode that becomes playable after finish story mode once. (You don't have to play "Normal", because it has done when complete a story mode playthrough for the first time)
Basically, you can choose any main game mode, but since "Boss Rush" doesn't exist in the "Story Mode", need to play it in the "Single-Player Mode" or "Co-Op Mode".


Note:
[*1] The secret achievement "Storyteller" is also unlocked together.
[*2] This costume is necessary to get the "Bun! Bun!" achievement.
[*3] The secret achievement "Dummy Hater" is also unlocked simultaneously.
[*4] This costume is necessary to get the "Quack! Quack!" achievement.