I think I’ll get an estimate from a tool and die repair place.
You might think a new stick is not so bad a deal, once you get a quote from a machine shop !
You could just take whatever knob you're going to use, bore it out, put a 5/16" insert in it, and call it done.
OR, like Triley41395 said, weld it up, and re-tap the threads to 3/8" - But I don't think you could cut them with a lathe, as the stick is probably curved - You'd have to tap it.
It could be welded (or brazed) with TIG, and with a heat sink applied below the weld, you'd probably be able to save the chrome.
OR, the stick could be ground down and shortened a bit, if you could live with that, and new threads cut from where the good 3/8" ones start.
If you were careful, it probably wouldn't be noticeable, just a bit shorter.