1950s Theo A Kochs Pump Handle Bolt Size

You probably have the wrong threads...I have a nut and bolt thread checker set that hangs in my shop, everybody should have one, sure saves time...the common thread sizes are 24 and 32 for SAE

I also have a tap and die set for clearing threads when they've been damaged...also a lifesaver...

My thought exactly.

What would the bolt (actually cap screw) hold?

In theory it would be some kind of handle cap. Some of the pictures I have found, it was a ball end, others it’s like a beer tap handle for lack of better description.

Having an end on the main pump handle would make it more comfortable in the hand as well as give a little better leverage to pump the chair up.

based upon the age there are four pitches for the 3/8 Bolts: 16, 20, 24, & 32. rather then a round thread gauge, get a flat style that is not diameter dependent

Theo H Kochs was founded in the 1870's and as such, there was no such thing as a standard thread. This is even prior to the Philadelphia Convention standard threads which later evolved into United National thread series. You may get lucky and it will turn out to work with 3/8-24. I worked for a company founded in 1902 and there were several weird threads, like 9/16-27 and 13/16-8 that were originally used and maintained to keep interchangeability of parts. If the chair was built before WW2, the likelihood of weird threads increases sharply.

If 3/8-24 doesn't work, you will need to measure the thread pitch and diameter and very probably get a stud cut to that thread. You can get a short piece of wooden dowel that is a snug fit on the minor diameter, thread it in a few turns and measure the protrusion of the dowel. Then turn the dowel in 5 turns and re-measure the protrusion to get the thread pitch. From the indents on the dowel OD you can get a minor diameter for the thread as a starting point.

...or you could drill it out and install an insert....

Depends on whether you have a commitment to maintain the original equipment as a conservator would or if you just want to get it running. In some circles, the insert would cause lynch mobs to form and crowds with pitchforks and torches to gather.

Easier yet would be a handle shaped object with a smooth shaft and a gob of JB Weld, or a smooth shaft that fit the hole, a bigger gob of JB Weld, a ziplock bag and a rasp.

Well antique barber chairs are worth more restored, there seems to be no bonus paid for originality....A nice looking functional chair is the goal...