Or...
The blind rivet/rivnut is going to pull out for the same reason the sheet metal screw did, won't it? You still need to spread the load.
I can think of two ways to get your washers/doubler installed, neither of which is pretty:
1) The bed is just a huge box bolted to the truck frame, right? So, unbolt it, shift it, install your hardware, and put it back together.
2) Go into the cab, remove the stuff against the back wall, drill holes big enough to do your installation, then permanently plug the holes.
I'd say both of those fall into the Not Worth It category.
Or....(I think you're going to like this one)...
Here's an alternative way of preventing overstress on the screws:
The reason your screws are pulling out in the first place is because the box is trying to move away from the position you've got it in, right? So...give it a way to resist that without putting a big tension load on those screws.
Use angle brackets screwed to the bed to keep the box from wanting to move. Make a frame of 1x1 alu angle or equivalent the same size as the box footprint (with the flanges outward so the box can sit flat on the truck bed) and attach the frame to the truck bed. Set the box down inside the frame. Now the box can't shift laterally anymore...in fact, you could probably drive around like that without even attaching it and it wouldn't move unless you hit a big bump. You don't even need a full frame...two "chocks" per side (front and side(s), since the back of the box is screwed to the truck cargo box sidewall) should do it.
You can use screws there because the load is in shear. Since the box is positively located, it won't be trying to pull the screws you've got going into the side of the bed out all the time. Properly prime/paint or otherwise corrosion-proof the bed floor brackets and screws to make sure they last.