Don't bother riviting, welding is the best way. You just have to find ways to control the burn through.
When I did my CJ-7 it had rust in the same places, under the roll bar and in the foot wells.
I made my own panels and used a flanger to make lap joints instead of butt joints.
The lap joint helped with the burn through problem but if I were to do it again I would use butt joints instead.
The problem with the lap joint is getting them sealed afterwards. If you can't seal them up they will rust out.
If you look at some of the work the Hot Rod people do you will see they tend to use tight butt joints.
They clamp everything in position then tack weld it and then fill in using short welds. Let things cool down as you stitch things up or the heat will twist things out of shape.
Another thing that I've seen used is these little panel clamp blocks that hold the two pieces tight together for a good butt joint.
As far as controlling burn through there are a number of things to consider.
First, if there is any rust pits the metal will burn through, there just isn't enought metal left. Cut back the metal till you have clean metal and use a bigger patch panel.
Use the smallest diameter wire you can get for your mig. The smallest I have is 025. The wire diameter controls the heat of the weld almost as much as the settings do. The opposite is true when you are welding thick material with the mig, use the thickest wire you can.
MIG's are CV machines (constant voltage), so like the flow of liquid through a pipe under constant pressure the cross section and length of the arc determine the heat.
Also, weld in short strips, as the surrounding metal heats up its more likely to burn through.
If the metal warps out of position it will also want to burn through.
If the back side is open you can try using a brass or copper backing block. The block supports the weld and also absorbs some of the heat and the steel wont stick to the copper.
On my next project I think I will try making a panel block and backing block combination.
The following pictures are of the flanger I used and a test butt joint I did recently. The weld penatrates nicely without burning through.