Field report from the back yard: The skewers work just fine, and 1/8" diameter rod is not too large, and they are stiff enough that they don't flop when being handled with the tongs. The smooth surface makes removal of cooked meat easier.
We also found online suppliers of the better grade of skewers.
I think I'll throw the old plated rusty steel skewers away.
I also found the following posting, which pretty well summarizes what's needed for welding, hard soldering, and cleaning the firescale off stainless steel: .
And these guys are forging Type 304. Apparently electropolishing will restore the surface. And there is one poster who used to make food service equipment, and he said that Type 316 better resists cleaning solutions than Type 303/304.
So, we have the what-material-is-best answer: Use Type 303L, 304L, or 316L.
Joe Gwinn