Hello Stuart,
Planning a family holiday to Walt Disney World Resort does indeed take a lot of thought and consideration especially as to when is the best time to visit and you've already got 2 great months in mind!
British visitors like us fair better in the more temperate Florida winter or spring weather and generally these months are not too over crowded either. However it is always worth avoiding US National Holidays such as President's Day on 15th February 2016 and the Easter holiday period at the end of March 2016.
In February Walt Disney World Resort typically hosts the hugely popular
Disney Princess Half Marathon Weekend which attracts large crowds; although I can't say for sure if or when it will be held in 2016.
April usually includes
Epcot International Flower and Garden Festival which draws the crowds but because this event is held over several weeks it shouldn't really impact too much on a 2 week stay.
As for accommodation suited to your family, you may wish to consider
The Cabins at Disney's Fort Wilderness Resort; me and my family of 6 just love them! The Cabins provide 1 x bedroom which comfortably sleeps 4 in 1 x double bed and 1 x bunk bed plus 1 x double sized pull-down bed suitable for 2 adults in the living area. The peaceful quiet Resort of Fort Wilderness is only a short boat ride away from the razzmatazz of the Magic Kingdom Park.
I'm sure whatever is finally decided upon you'll enjoy a magical holiday to Walt Disney World Resort making memories with your dear family which will last an entire lifetime.
~ Jane