I fly and camp, and definitely don't use backpacking stuff.
On most airlines you are allowed two 50lb check-in bags/boxes per person not exceeding 62" (HxWxL), one carry-on, and one personal item.
(On Southwest Air, you are allowed three 50lb check-in bags/boxes per person)
With my allowance for one person, I bring the following:
Plastic Footlocker
Large Duffle Bag
20" RollerBag
Bookbag Size Backpack
13x13 Coleman Tent
9x11 Rain Shelter
1 Folding Chair
Air Mattress/Pump
Electric Blanket
Pillow/Sheets
Battery Lantern
Small Fan
Stool
3 Changes of Clothes
2 Towels
2 Washcloths
Digital Camera/Docking Station
Laptop
Cell Phone
40 ft Outdoor Extension Cord
Power Strip
Laundry Detergent
Dryer Sheets
3 Rolls of Quarters
Extra Batteries
Toiletries
I put anything that is small, but has some weight to it in my carryons such as my Lantern, Batteries, Air Mattress Pump, Extension Cords, Quarter Rolls, Batteries.
Anything that is large goes in the checked luggage.
I don't cook while at Ft. Wilderness, so I don't have any of that.
From your picture it looks like there might be three of you, so with 6 50lb checked bags, 3 carryons, and 3 personal bags, you should be able to manage without shipping anything.
I don't think a gate-checked stroller counts in your luggage allowance.
If you fly Southwest, you can have 9 50lb checked bags, 3 carryons, and 3 personal bags.
The only thing you can't transport on the plane is propane or lighters, you would have to purchase that at Ft. Wilderness.
Also, if you are renting a car, a quick trip to Wal-Mart to purchase what you need then dispose of it will probably be cheaper than shipping.
If you are not renting a car, then Magical Express will take care of all your stuff.
I hope this helps.
Carol