Not only will it not cost you more to rent offsite, it will actually cost you far, far less and you will have so many more amenities.
We only have 2 kids, and we rent 2bdrm condos, minimum. I like having the space for everyone to get a proper night's sleep. I like having more than one bathroom, and I love having a full kitchen to have coffee in the morning and make a light breakfast and snacks. I love having a fridge to be able to stock up on water and juice to take with us to the parks, and to keep the beer cold so we can have a nice drink on the screened patio or balcony. I love having a washer and dryer in unit so I can bring home clean clothes and not have to start doing laundry the minute I get home.
All of this will cost you approximately 1/2 of the price of staying at Port Orleans. We are getting quotes now for our Dec stay, and I have prices ranging from $399 for a 3bdrm condo for a week. I am looking exclusively at Windsor Hills as it gets the best reviews from the folks here, and it is less than 5 mins to Animal Kingdom. I received a quote today from a lady with a 3 bdrm townhouse, with it's own splash pool and the price is less than $800 for 8 nights, and that includes all taxes and fees.
With 3 kids, I am certain that just having the kitchen to make breakfast will save you another ton of money. I can't imagine if I had to get my family out the door every morning to a restaurant or food court. We would be lucky to get to a park by 10:00 am! Getting 5 people out the door with one bathroom to get ready in would really suck.