Hi Jennifer!
First, you are very lucky that your child will nap so easily! Neither of my boys were great at this, which meant we always had to go back to our resort during the day for them to rest while at Disney World.
I think you should check out the
restaurants at Downtown Disney. There are so many that you are bound to find one you like. You don't need a ticket to go to Downtown Disney, and you could stroll around with your napping child after you eat. My favorites there are Portobello, Paradiso 37, and Raglan Road. Portobello has great pasta, which would be wonderful for your race. Just be sure to get a light sauce on it, nothing too creamy or heavy.
I would make the reservations for around 1:00 p.m. That will give you time to finish the kids races, head to Downtown Disney, and still have time for the food to digest and you to rest a while before you go to your race.
I hope this helps! Please let me know if you have any other questions.