Hoot,
You didn't post photos of your rifle, including proofmarks. This might show bore(not groove or bullet) dia.and case length.Sauer and Sohn chambered for a cartridge very similar to 9.3x57R/360. This cartridge is 9x58 1/2R S&S and is very often confused with the 9.3x57R.I couldn't find where you slugged the barrel.If you haven't done so, you should. Old 9.3s are found with groove diameters of from .352" to .368" (the jacketed bullet you used is the standard 9.3x72R bullet and is made to be safe in any of them).If the barrel has an uneven number of grooves,you may have a little trouble measuring the slug,because the grooves in the barrel are not opposite each other.For cast bullets, I prefer a bullet about .002" over groove diameter.BTW, when I use the term groove diameter, I mean the larger diameter, and for bore diameter, I mean the diameter before the grooves are cut. I only mention this because SKB shows it the other way, and I don't want to confuse matters.I have no experience with paper patching bullets, so will leave their diameter to SKB and Sharps 4590. If your rifle is not nitro proofed, you can load nitro for black with good results, although some black powder rifles may be more accurate with black powder and proper bullets and lube.
Mike