I would get all the camping gear out and pack it in the truck. Get out the (empty) suitcases, duffles, camp chairs, put it in the truck. Try to pack up your truck with everything you plan to take (using empty bags/totes to represent full ones). See how much room there is.
If you plan to put totes in the back of an uncovered truck, be sure to duct tape the lids. They can and will come off going down the highway.
I think I would take my own car, bc I know it. Also, you said it has a cap on the back. If you rent a truck without a cap, your stuff is open to the weather and also other people at every rest stop and restaurant. If you stay at a hotel, will you be comfortable leaving all that stuff in the truck bed? Or else your hotel room will be full of your camp gear and everything else you don't need.
With three kids, I don't think a small car, or even a large car, will fit everything you will need for both a cross country trip AND camping. Last year we had a mid sized SUV, with me and my two kids- when we went camping, we were PACKED in there. No room for wiggling. We were only going about 2 1/2 hrs away. Any further, I would've left a kid on the side of the road.
If I were you and were to rent a vehicle, it would be a minivan and I'd fold down/remove half the last row, since you will need room for 5 when your mom is with you. But I think adding camping gear will still make it a tight fit.
Good luck!