Thanks for the copies of the critical pages from the 1936 and 1937 Western catalogs. The K####T numbers were in the 1037 catalog but not the 1936.
These pretty much confirm that the late 1936-early 1937 as the introduction of the K####T numbers at Western. The only question that is outstanding is that there is a Western 380 “Target” box with a load date of Nov 1935 that has a Western K# (K1354T). One possible explanation is that the box has an “A” prefix on the load date code so it was actually produced at Winchester. It may be that Winchester assigned these numbers for their production of Western Cartridges, but the codes were not picked up by Western until 1936/1937. The only kink in this theory is that there is another box, a 9mm P export contract box-not a Target box, with a load date of May 1935 that doesn’t have a K####T code on it.
Thanks for the great info!