I won't try and talk you out of the U-Haul or into the PUP. What I will say, is that when I was (much) younger we used to go camping every year. We had a PUP (very VERY basic. 2 slide out beds and a bed in the middle that converts to a table. No Electric, water or gas). The 4 of us kids (2 brothers, 1 sister and me being the youngest son) slept in the camper. Mom and Dad slept in a Canvas similar to this one:
(but more colorful)
For our "Kitchen/Eating" area we had a Shelter (about 12x12) with a canvas roof similar to this:
Ours was NOT the easy up style, rather 4 corner poles, a center pole and ropes and stakes around the outside to hold it all up. Again, CANVAS. Both the tent and shelter were heavy (as canvas seems to be). The original boxes had long been gone and my dad made wooden boxes (plywood) to hold each with it's poles, ropes, stakes and canvas that fit into our PUP. All of our "camp" stuff (coleman propane stove, lanterns, kitchen stuff, 5 gallon collapsible water jugs, first aid kit, etc....) all stayed in the camper and there was usually room for 2 larger coolers (1 with "Food" and the other with beverages) as well as some other food stuff. The rest went into the back of our Station Wagon. (mostly clothes and some other food stuff). Then the 6 of us (Mom, Dad, and us 4 kids) would load up to go camping. No problems towing it. Some of my best memories were these annual trips. The campground, however, was only about 10 miles away. Of course when you are a kid and excited, 10 miles might as well be 100.
We also (when I was even younger than that. Maybe about 2 years old) made a trip from Northeast PA to Colorado to visit family. We took the same camper and made stops along the way. Sadly, I don't recall any of that trip, except for pictures I have seen over the years.
So while the weight of the PUP could certainly be an issue, it may also work the other way. Whatever way you go, I hope you and your grandkids have an awesome time.