Do you have a chamber cast and have you slugged the bore? Also could you post a pic of the 2 seemly different rounds? If I remember correctly, the 8X57 IR typically had a 12.9/13 gram bullet. Originally the military version S2 had a weight of 10.3 grams and a diameter of 8.23 +/- 0.02mm and by 1904 the diameter was standardized to 8.22mm and a weight of 9.7 grams.
The longarm was sourced from Erdmann Schmidt & Eduard Habermann, E. Schmidt & Habermann, and is a Model 581 A&D(or a Model 580 if a Blitz action). It is a post 1912 and pre-1923 example which is a difficult period to decipher the "R.S." which could be for Robert Schlegelmilch who was ending his career or Richard Schuler who was just beginning.
Kind Regards,
Raimey
rse