Nah, Demon Hunter may have animals as partners, but the real class focus is all about speed and skill. Rainbow Dash was made for that role.
Fluttershy would probably be the Witch Doctor. That class is actually relying on others to do the real fighting, and since I can't really imagine her bring up front with enemies, that'd be my bet. Plus, some of the darts the W.D. shoots act more like a tranquilizer than a poison, and given her peaceful nature, it would really fit her character to hit enemies with rounds that might be consisted to be less lethal and more calming.
This class also mixed in her like of animals, but even better, her lack of fear for them. She's shown time and time again that she isn't afraid of beasts that may be big (within reason), or scary, or creepy. This would tie well with summoning giants (they could be re imagined top be bears or some other large mammals), undead dogs, and spiders, respectively.
Plus, it's not like Rainbow has an aversion to animals either. The Demon Hunter used animals that are fast, agile, and dangerous. Back in "May the Best Pet Win", before Tank saved her, RD was very much drawn to pets like these, to the point where she had to make a contest to decide which pet was the most 'awesome'. Back to the D.H. class, you have wolves, ferrets, bats, and most notably a raven as the signature of the class. Since, if it weren't for Tank proving himself right at the end, a hawk was going to win the final challenge of the competition, it is not a large jump to say that RD could be an using a raven as a companion.
--TL:DR VERSION-- Fluttershy makes a better Witch Doctor, and Rainbow Dash makes a better Demon Hunter.
Nah, Demon Hunter may have animals as partners, but the real class focus is all about speed and skil
