I always try to be more understanding with young kids when you are somewhere else than home. Maybe the pizza tasted differently than she was accustomed to and she didn't like the cheese? You even said it was super greasy! I'd say let her peel it off and toss it if she didn't like the grease.
Kids go through phases with food. When my daughter was small (like 3 years old) she would pick off all the parts and eat them individually. Bell peppers. Olives. Pepperoni. The cheese. And would leave the crust on her plate (which was fine by me, it just made it into a salad with cheese and zero carbs, lol). Then, she got where she picked everything off and ate only the crust and some of the cheese (if she was successful in picking everything else out of the cheese). Then she would eat pepperoni and black olive pizza (no other toppings). Now, she eats pretty much everything on pizza (onions included) except she doesn't care for mushrooms.
Just let it go. She's 5. It's not like you were at home and prepared it exactly the way she is accustomed to it and you were practicing manners. You were at a party. And even kids can be allowed to act like a kid at a party. It's a once in a while thing!