You will need airbeds - they take up less space than rolled up self-inflating or foam mats. A large picnic blanket will avoid the need to take a table and chairs, although we found something like this a revelation in terms of making sitting on the ground more comfortable: www.amazon.co.uk/Highlander-Outdoor-Seat-Blue/dp/B002WUJC5U/ref=sr_1_4?ie=UTF8&qid=1390412891&sr=8-4&keywords=lightweight+chair. If you want to sit up in the evenings drinking wine and reading, you might want to get one
.
If you go to a site which allows fires, you might be able to rent a grill so you can cook over a fire but, even then, you will need some sort of lightweight gas stove for boiling water and pasta and rice maybe.
Bring pillow cases and then you can all use clothes stuffed inside them as pillows.
Bring the minimum of utensils and crockery - lightweight mug, bowl and maybe a spork or a knife, fork and spoon for each person - cooking-wise most things can be done with a set of thongs and a fish slice or slotted spoon.
Don't forget lights - a head torch for each person is probably the most practical and most light-weight way of dealing with it, and maybe some sort of camping lamp for the tent if you can squeeze it in.
Some sites have fridges and freezers but, if not, a cool bag will be easier to pack than a cool box, through not as good. Freeze milk, bottles of water, bags of wine etc. before you go and these will defrost slowly and act as ice packs while you are away. Do as much of your food shopping as possible after you have arrived and unpacked - head off to a local supermarket and stock up there if possible - that will avoid packing too much food.
Other than that, clothes are going to be your biggest issue probably and you can avoid too many changes for the little ones by putting them in wellies and waterproof trousers (unless it is shorts weather obv) so it doesn't matter when they (invariably) fall over in mud/puddles.
Mind you, if you are only going 15 miles, your children can have lots of stuff packed beneath and around them which will hopefully provide a bit more space.