I have a solution!
Since you would like to dine at a character breakfast in the Magic Kingdom, reserve the latest seating possible at Crystal Palace. Dining with Winnie the Pooh and friends will be a perfect backdrop to your son's birthday festivities.
A late seating will be similar to a brunch rather than lunch. Yes, you'll be having breakfast food but by the time you are ready for cake, it'll be near 11:00 a.m. Or, book the earliest lunch reservation so that you can dine on lunch items and then have cake.
While I think you could bring in a cake to eat at an outdoor location (since I don't see cake listed as prohibited), carrying in a birthday cake to the Magic Kingdom could be more stressful than a vacation deserves.
There's lots of ideas for birthday celebrations
here, including info on how to order birthday cakes!
Another idea would be to have your breakfast at the Magic Kingdom, tour the park, then return to your hotel for a mid-day break. During the break, enjoy cake at your resort. It would be much easier than trying to get a cake to a location in the Magic Kingdom.
I hope your son has a very special first birthday!