The truck may not be the favorite family hauler but will pull your camper with a lot less stress and more safety. I still don't understand why Discount Tire couldn't come up with load range E 20" tires for your family vehicle. Here are a ton of 20" LRE tires and most a product lines Discount Tires carry.
https://tires.tirerack.com/tires/275%2065%2020
Just be thankful you have a family that is enthusiastic about camping and take the time to get the kiddos off their electronic brain killers and have them out and about getting some scrapes and stings as this stuff makes for better memories than playing Nintendo or whatever they play these days (probably is Pong I suspect).
BTW: You a firefighter? I was with the Mico VFD for 12 years and ended up an engineer as I got a little to... well, husky to get around well in turn out gear. Lucky we mostly do wildland firefighting and I quit/retired when I turned 65 as that got a bit much for me.
I know your family would have been happier going to a campground in the Nissan family truckster but they will be a lot safer in the real pickup. You will also have a lot less chance of burning up your Nissan brakes and transmission and ruining the suspension.