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So when I was reading ROTG fanfics for the first time, I got confused when they referred to Bunny as a pooka because I hadn't read the books. So I looked up what a pooka was and ended up getting the Celtic shapeshifters instead before I got my search words sorted and figured out that Bunny was an alien. But then I came across this:
According to legend, the púca is a deft shapeshifter, capable of assuming a variety of terrifying or pleasing forms. It can take a human form, but will often have animal features, such as ears or a tail.[10] As an animal, the púca will most commonly appear as a horse, cat, rabbit, raven, fox, wolf, goat, goblin, or dog. No matter what shape the púca takes, its fur is almost always dark. It most commonly takes the form of a sleek black horse with a flowing mane and luminescent golden eyes.
Does that last description sound familiar to you?
It sounds just like the nightmares.
So we're fudging the timeline a bit. Pitch created the nightmares a long time ago, before the dark ages even. However, some of the nightmares didn't like working for Pitch and ran away from him, going into hiding, eventually inspiring the Celtic myths of pooka shapeshifters, with Bunny being one of them.
But Pitch eventually found them and either took control of them again or just killed them all like he did with the alien pooka in the books leaving Bunny to be the sole survivor, not that Pitch knew that at the time.
Eventually Bunny got chosen to be a guardian and he pretended he was never a nightmare or a pooka, just sticking to a giant humanoid rabbit form as his default and ignoring all of his shapeshifting powers because he's in denial. Oh, and he might've decided to be Australian just to throw people off and help with the whole disguise thing.
Then the movie happens and Pitch senses Bunny's fear that if kids stop believing in him and he'll get magically depowered then his disguise will fail and everyone will find out he used to be a nightmare.
Cue Pitch exposing Bunny and trying to regain control of his 'runaway minion'.
Also, Sandy probably knew that Bunny was a nightmare the whole time and kept his secret for him because he's nice like that.