The budget board has some great tips. Here is some of what we did to save. Split meals with our then almost 4 yo. Shes a light eater and never finishes a kids meal. Ordering a separate meal for her was a waste. I ordered her an extra side and just fed her off of my plate. Also, skip sodas. We drank water mostly. It was hot (may) when we were there, soda was not appealing to us at all.
Do late lunches for your ts meals and eat a lighter dinner. Ts lunch is much cheaper than dinner. Or schedule a later bfast, then have a light snack for lunch, then a decent dinner.
You can order groceries from a delivery service or make a grocery run if you have a car. Breakfast in the room is a great idea- saves both time and money. Cereal, oatmeal packets, yogurt, things like that. Since you have a fridge, this should be very doable.
For us, it really wasn't too hard to stay in a budget at Disney. We had some nice ts meals, we did liberty tree lunch, crystal palace lunch, and akersus bfast. We still enjoyed ourselves.
We don't enjoy eating big, heavy meals or desserts after being in the heat of the parks, so I am sure the fact that we naturally leaned towards salads, sandwiches, etc helped us stay on budget.