So I have done the math involved, which is absolutely none, and figured out what classes pinkie can be. I will now list all classes and why she can or can't be it.
Scout: When she wants to she can go extremely fast. Also jumping like there is no tomorrow.
Soldier: Crazy...just crazy.
Pyro: Craziness that can get real dark real fast.
Demo: See soldier, just a bit less.
Heavy: Sandvich. Also other food. Also much smarter than initial appearances lets on.
Engie: She builds many much things.
Medic: This one is a crap shoot. She has the ability to make others happy (kinda heal), but she also isn't really the best at physical healing.
Sniper: Probably not. Usually she doesn't have the patience for waiting around but, if we take into account the episode "Too Many Pinkie Pies," she is patient when she needs to be.
Spy: This one is either/or. If we go by her ability to hide in plain sight on the episode "Party of One," she sucks. However if we go on the same ability in the episode "The Crystal Empire: Part 1," she's absolutely amazing.