I decided to solder this joint. My soldering iron was buried somewhere in my workshop, but I had a MAP gas torch and plumbing flux and lead free solder handy, so I went for that. I knew it would be risky with respect to burning the paint, but I went for it. I did burn the paint. If you want to use a torch to do the same thing, turn that flame down low. Also, be sure to have your lunch box oriented so that your solder does not run, but pools in the area you want it to be.
If I were going to secure a metal lunch box handle again, I might try JB Weld. If I were going to solder it again (which I might, if I don't have JB Weld around), I would for sure dig out my soldering iron to have more localized heating than the torch.
Overall, it is more robust than it was and I don't expect to be messing with it again.