I would try and find some of his fired cases among his equipment, if it is still available. Lacking that, the chamber cast is the way to go.
Common practice was to indicate the bullet/bore first, then the case.
If this happens to be correct, the cartridge may be based on a 338 case, with .375 bullets.
Could the Smith have reversed it? Maybe?
Suggest you cast the chamber and make certain.
As others mentioned, measuring the bolt face should give you some useful information about the cartridge.
I did find some data for a .375-338, referred to as a ".375 Belted Newton" and ".375-338 Chatfield-Taylor" developed by R.F Chatfield-Taylor of Manchester, MA. (PO Ackley's Handbook for Shooters and Reloaders, Volume II, page 212 and 213). Load data indicates 300gr bullet velocity of about 2500fps, test barrel length was not stated.