Just FYI, you won't be able to throw a blanket down and have a picnic inside the park. You can bring in food and sit on a bench or on the street curb and be discreet, but they do not want families having a full picnic with food from outside WDW (and you won't be able to use the tables at the counter service restaurants either). So keep that in mind...I was wondering how you would get the food and transport it over the park etc. without spoiling too.
If you are looking for a chicken bucket at a reasonable price, you could take the ferry over to Fort Wilderness and they sell a family chicken bucket meal at Trail's End. Or find something easier to bring like snacks and PB&J.