Here's my repair. Used a bit of 3/4" stainless (not sure which alloy; it was an estate sale find, but it machines really nicely). Chucked it up, turned a portion to about 0.5", and threaded M6 1.0:
Put the cross-slide at 45deg and did the shoulder, which resulted in this:
Chucked it up the other around, indicating for center (my 3-jaw doesn't have reversible jaws, so working close to the middle is a pain, hence the 4-jaw):
and rounded the end in stages, with cuts at various angles obtained by angling the cross-slide, with some filing at the end. Now I have this:
Put it back in the soon-to-be-less-annyoing vise, and milled about 90thou off each side:
and made a pin from 1/4" drill rod. Oddly, the holes in the vise are just under 1/4" (maybe distortion from heat treatment?) so I had to turn down the ends of the pin a little.
I then drilled and undersize-reamed a 1/4" hole, and pressed the pin in:
It then took some fiddling to get it to fit the vise smoothly; the slot in the vise actually wasn't centered, so the pin has to stick out one side a bit more, and I had to stone down the vise slot at one end so this part can slide all the way to the end. Boy the vise metal is hard!
It seems quite a bit better, but time will tell if it solves all the problems.