I did a quick search of camper rental places that will set up at Fort Wilderness and found that there are units available that will sleep up to 10 meaning you have extra beds even if they are doubles. Just look at the floor plans or even call them and ask how the sleeping arrangements are set up.
Here is what I would do in your situation.
Rent a travel trailer or 5th wheel that was set up for me and pack some dry goods in a spare suit case to bring with you. Use the suit case for bringing home souviners. Use these for breakfast (granola bars, breakfast bars, milk and cereal bars, shelf stable milk if you can stand those (I have one child who for years drank those when we would go camping because said child wanted their milk so much and there really wasn't room in the ice box-when you have to put old milk jugs filled with ice to keep things semi cold it is an ice box not a fridge so only things like butter, cheese, sandwich meat, eggs were able to be kept in there), snacks at the camp ground or even taking to the park each day as extra snacks (nuts, jerky, tuna lunch kits or even just the tuna packets , favorite candy personally I am partial to going to a store in my community where I can select exactly what flavors of Jelly Belly I want. That and a big bag of gummy bears)
If the company that you rent from stocks the fridge for you then you could easily get things like milk, eggs, cheese, sandwich items, ect or you could do a ride share to a nearby store off property to get those items as well as getting things that don't necessarily require a fridge but things that won't fly very well because they would get squashed like bread.
Even if you did use a dining plan (going with the regular for this example) you could have this set up
breakfast at the camp site using items you have on hand
mid morning snack in the park using a snack credit if you have some almonds or other nuts with you could easily get something like a cut up fruit selection and add those for some protein to help keep you feeling full longer
counter service snack in the park using a counter service credit
mid afternoon treat in the park using a snack credit
table service credit at your choice using a table service credit or 2 for a signature place
snacks back at the camp site in the evening or even while waiting for the night time show or parade.
If you didn't use the dining plan you could take snack items to the park, and plan on having one other meal each day back at your camp site while you take a mid day break or at the end of the day. Do get a special treat once in a while though such as a dole whip.