With a roller feed rates will be a big deal. I imagine it would be hard to do freehand. You probably could find a way to tig it pretty easily.The roller allows the heat to concentrate under that tiny point which reduces the amperage to a usable level.
I know a guy who can weld two Gillette on edges.....If you can TIG HT foil I bow to your welding skills.
Might be able to do it if you clamped a strip of copper on each side and left the foil stick up just a hair 1/16. And then just fuse the seam in a quick pass. But still folding would be faster at that pointIf you can TIG HT foil I bow to your welding skills.