I'm a little confused about one thing in your diagram. Why are you running your ground lead as a positive and your stinger negative? Wont that create a short circuit and arc as soon as you connect the ground clamp to your rig? I could see that being ok for welding on metal not touching your rig but what do you do if you need to repair your own vehicle?
Correct "Ground" for Flux core welding is Positive, and Negative is the stinger.
I tried switching the polarity to see what happens, it just spits metal balls everywhere, nothing close to resembling a weld.
When welding on your rig, you need to disconenct the vehicle's entire electrical system. (a good practice even when at home) I take the battery out of the vehicle and set it on the ground and borrow an additional battery from someone else in the group. Then I put a blanket over them incase something bad happens and they decide to explode. I think the explosion hazard is less than with stick welding as the current transfered by the wire is less, and the direct short possibilities are smaller. The batteries are 100% isolated from the vehicle's electrical system.
If I had 2 batteries in my vehicle, I would leave them in place, but still disconnect the cables, convert to 24v, and weld up.
Just to be clear the goal of this project was to have a cheap trail mig that was reasonably portable. Cheap being the most important factor. In the orange tool box i use. I have the jumper cables, welding goggles, 15ft extension cord, 4.5 grinder, gloves and goggles. The tool box fits nicely on the floor behind my passenger seat in "unused" territory.
As far as the briefcase welder, (Ready welder is the brand) is designed for trail use, exactly like this. The street price for the base model on that is $500 and goes up from there. I also did not like the spool gun as I felt it would be hard to get that moved around to some of the un-ideal conditions we sometimes need to weld in. The regular gun can get in tighter spaces.