From my experience, kids are actually really hungry at that time of the day. That day they will eat lunch, take a nap, wake up, go to a party and exercise, and will want to eat something. Don't you think? Maybe that's just my kids, but they eat five smallish meals a day instead of three huge ones. I'm sure their parents will give them something to eat after their nap times, but I would be prepared to have a few hungry kids.
Other ideas besides a pizza meal, would be to go for a bunch of little snacks, like appetizers and finger foods that the parents can choose from for their kids. At our last party (4 year old), I went to Chick Fil A and got 12 packs of their nuggets. I think it was $4 for each pack and they gave me plenty of dipping sauce. Those are great even when they're not warm.
I would also bring a veggie and fruit assortment(lots of dairy/wheat/gluten allergies out there).
Be prepared for the kids to want two drinks and for the parents to want one drink. That's usually what happens at our parties. You don't want to run out of drinks.
Also, you might want to pack a few rice krispie treats for the kids allergic to cake. That seems to be a hit with those allergy moms at parties.
Have a great time! Both of those places are good fun and you won't want to waste too much time eating!