Good question. Bring as much as you want to put in your kitchen, beacuse there will be room for it all. What you don't want is a bunch of camping gear on the bed that has to be pulled out every night.
Don't bring:
Chaise lounge chairs (perhaps bring a pair of lightweight chairs)
Portable fire pit
privacy tent
portapotty (unless you are really going to be roughing it)
10 x 10 pop-up tent - bring the tailgator if you have it
extra tables
bicycles
kayaks
We found that after a long day of driving, you just want to pull over and climb in bed. As our trip went on, we spent less time looking for campsites and more nights at the nearest reststop. Of course, the weather was in the 20s, so we had zero reason to want to be outdoors. In your case you may want a few outdoor "camping gear" items - but try to keep it as light as you can.
One other tip is pack a suitcase in the back of your tow vehicle, and each night grab the clothes for the next morning and put them in the C-I.