Camp chairs. Get a comfy one. After being in the saddle all day you don't want one that cuts into the back of your thighs or hurts your lower back.
A lot of what you bring should be determined by whether you will have a "base" camp or will camp/ride/camp/ride. From your post, I am assuming it will be more of a base camp situation.
With that in mind, if it fits in the truck/trailer, take it. Especially the first couple of trips. If you find you got along without something, leave it home. That's the beauty of the bins.
Use Word or Excel if you have them to create a checklist of your camping gear if you have to take stuff from your home or barn. This way you remember what to grab as you are packing. (If you have an iPhone or iPad there are some great apps for that).
A lot of what you pack will be dependent on the location, amenities (rustic or modern campsites) duration of trip, and what you expect to do and how you will spend your time. The whole idea of "it's camping, I'll just eat peanut butter" is for the birds.
Camping does not mean eating cold food on a napkin. It can, but it doesn't have to. I have made great roasts while camping (using a cast iron kettle and fire pit). Don't forget to think about what you will do once your horse is up for the night and it is only 6pm. There is still a whole lot of daylight. Book? Portable DVD player? Crossword puzzles. Fire and glass of wine?
Enjoy your time off your horse as much as you do on your horse.
Hey, where are you located? This sounds like FUN! Hahahaa