How to Complete Fluffy Flier?

The objective of this quest is to defeat all foes. The foes are a Fire Drake, which is weak to Ice; a Banshee, which is weak to Dark; a Sprite, which is also weak to Dark; and a Brown Chocobo, which is weak to Holy, Water, and Lightning.

As the battle goes on, the volcano will erupt every few rounds, damaging everybody.

Feel free to have a Chocobo Knight catch the Brown Chocobo.