The location of this badge can be quite tricky. It is only obtainable after Mario gets the ability to transform into a straight line of paper and can go through thin holes. With this new ability, Mario can by-pass the metal bars in prison cells in one of the rooms in Hooktail Castle. Inside one of these cages is this very badge.
Hooktail hates crickets more than anything else in the world according to Kolorado's deceased father. With this info and this badge, Mario can lower Hooktail's defense by one and attack by two after Mario attacks.