I've been thinking about your build. I'm not jealous of your situation. I can see arguments for both scenarios. On the one hand, the U-haul truck already sports the box, and you're right about bolting the 4x4 hardware to it. It probably won't be that bad of a job. I've considered buying a two wheel drive dually with a crew cab and ordering a SRK and using my spare Dana 60 from my donor truck to build a swanky OBS hauler. I've read about the conversion from two to four wheel drive a bunch. Once you have the parts it can be done in a day. I know your worried about having enough GVWR to be legal and the box truck is gonna really help that out since it's got a higher one. That might be all the reason you need to keep the box truck right there. However, the crew cab is already a capable off-road rig. The frame can be dealt with in several ways. I once put a flat bed on a early nineties dodge D-350. It did not have a flat frame either. My solution was to lay a 8' long piece of channel next to the frame and scribe out the shape of the frame onto the channel. Then I cut that out of the channel and welded it on top of the frame. This made a flat frame from the back of the cab to the end of the chassis. I put the lip up, so I had a nice shelf for the bed to ride on. I basically built a new top for the frame. It was no taller at its highest point than the highest part of the hump in the chassis over the rear axle. I welded every inch, inside and out of the channel to the chassis. It wasn't that big of a job. Added maybe half a day to the whole project. To this day that truck packs tractors and bulldozers up and down the road. That was 15 years ago. When I put the flat on my work truck I simply used the proper sized spacers to balance it on the chassis and straps to attach it, but I wasn't overly concerned with flex tearing it apart.
So, you can overcome the crew cab chassis. If you create a flat chassis it'll also add some strength too. Granted it will still flex. And big rigs are C-channel front to back. And they flex TONS with no issue. We currently operates several big trucks in our feed business and we are off road a lot. We run full lockers in all axels so our trucks are extremely capable in tricky terrain. Most of these trucks use a single point mount in the rear of the cab and two at the floor pan. One on each side. While the front two are hinged and the rear is an air bag you could fairly easily change your crew cab to a single mount in the center of the cab. There's a lot of metal back there and it wouldn't take a lot of plating, or bracing to beef it up. With the bed off you can access the area fairly easily.
This is a mount I fabricated for a Mack that I took the sleeper off of.
This is a factory mount with a pan hard bar on an International 9200.
And any time I see a big truck chassis I'm reminded of my old Ford trucks. They are a lot alike. I also think you probably won't be jumping and pounding the crap out of your old crew cab as much once you get the box on.....this could help a little
! I've packed 20,000 pounds plus with both of my crew cabs. I know several other guys that have too. The frames are stronger than the flexing and twisting would lead you to believe. I've also mounted really long tanks to some big trucks, like 20' long and used the plates and springs to allow for flex. It works. I've currently got a M-923 military 5-ton with a 14' flat and a two thousand gallon tank strapped on it. I'm pretty sure it uses bolts and springs too to hold the bed to the truck. All factory. And it REALLY flexes when I go off in the fields.
I actually rolled this truck last summer, it's a 932. Very fun.
Any way. I wouldn't want to have to make your decision. But for me personally, I'd probably put the box on the crew cab truck. However, you certainly ain't going wrong by keeping it on the reg cab and swapping the 4x4stuff either. At the end of the day you'll do what works best for you. None of us posting here have to drive it, or live with it. Good luck.