So, on to the valve.

The valve assembly was pushed out of the tube.

The valve assembly, disassembled.

I used a short dowel pin to press the valve stem out.

The disassembled valve.

Using the directions in
Airgun Hobby Vol. 2, Number 3, on page 43, in an article titled "How To Make and Install Your Own Valve Seals" by Peter Ruut, I stamped out a 5/16" diameter disc from 1/8" thick 95 durometer urethane rubber sheet.

However I found that stamping the disc made the edge of the disc concave. This didn't seem ideal to me. I'm not sure why this happened, but will chalk it up to differences in material supply, ambient air temperature and possibly my use of an arbor press instead of a hammer...

So I turned down some 95 durometer rod to 5/16" diameter.

And drilled the hole for the stem.

I clamped a utility knife blade to the toolpost and cut the face off.

Then cut it to length. This worked well and gave a very smooth surface. I think I'll make a dedicated utility knife holder for the lathe.

The new seal on the right, old on the left.

I pressed the valve back together. Notice the bolt in the picture with the cutouts for the cocking pins.
I assembled the rifle and it held gas! Two days later and it's still holding fine, so I'm calling this repair good. I did replace the other o-rings as well, which certainly didn't hurt.