Well chicks are associated with Easter, chickens aren't, so they're probably some sort of special magical type that stays a chick.
Why was the plot here even started? The ending shows A) that it's possible for non-rabbits to be the Easter Bunny and B) that the title of the Easter Bunny can be shared. Why not then just LET THE BIRD SHARE POWER WITH E.B.? Yes he seemingly had that whole worm problem but the general idea of providing alternate snacks was a good one.
Carlos didn't directly asked, but strongly implied it, telling the Easter Bunny that maybe a chick could do the job of delivering the Easter candy. The Easter Bunny treats this like a joke and laughs, which, yeah, makes him a fantastic racist.
Why does Fred not wonder why a large rabbit he just crashed into is wearing clothes, but when the rabbit speaks he goes all crazy?
Maybe he thought it just happened to be big (it wasn't that big really) and it had run away from a kid who put clothes on it? If movies are anything to go by, people dress their pets up.
If taking the scepter made Carlos change into a bunny/chick hybrid, why didn't Fred change into a bunny/human hyrid once he became Easter Bunny?
I think it's because the bunny metamorphosis shtick doesn't work with humans.
It's possible that since he was sharing the title with EB, the rabbit quota was met, and it didn't need to transform him.
Also it was an official coronation ceremony, not just someone making off with the scepter. No idea what difference that would make, but that did happen.