Perhaps find as many backplates and chucks that are that thread size and try them. If you get more than one that's tight I'd adjust the spindle.The lathe was made sometime between 1921 and 1937, so I’m not too surprised if the dimensions don’t match current standards. It was pretty common back then for manufacturers to use threads that were off just a titch so customers had to buy OEM parts and competitors parts wouldn’t fit.
I think I’ll research doing thread measurements and see if I can identify what I need to do to mate the 1937 spindle with 2026 back plate. I suspect I might end up making up a jig and hand-chasing the Accusize plate. It would just feel sacreligious to modify the Barnes spindle.

I have a leftover backing plate for a 4-jaw that was from a barnes #13. You are welcome to have it if you need it. It doesn't fit either of my current lathes so I have no need for it.Anyone have a 1-1/2”-8 TPI back plate that I can buy cheap or borrow? My Barnes lathe looks to have an 1-1/2”-8 spindle, but when I measure it seems to be over 1-1/2” o.d.. I need to get a new chuck for the lathe (the century-old Cushman chuck is beat-up and best I can get is 0.015” run-out) but I can’t justify spending a couple of hundred on a chuck that might not fit.
So if I can find a test back plate or chuck I can confirm if a current back plate will fit, then I can decide if it’s worth buying a new chuck.
Any options put there?
DM sent, thanksI have a leftover backing plate for a 4-jaw that was from a barnes #13. You are welcome to have it if you need it. It doesn't fit either of my current lathes so I have no need for it.