In my experience, the 7.8 mm notation on the barrel stamps of a prewar german or austrian rifle denote the 8X57 J caliber. if its a rimmed round (almost always for a drilling or combo gun) then is the .318 diameter 8X57JR cartridge.
The good news is you can fire a 8X57JR in the larger rifle with no worries. BUT NOT THE REVERSE. course you know that.
FOr the price, I would buy some Sellier bellot rounds, shoot into a sand bank or other recoverable medium, and then mike the bullet base. I believe you will find the grooves mike 7.8 mm and the lands mike 8mm at .318. if not stop and go slowly.
You can have a lot of fun developing both bullets and loads for the .318 JR round, if you purchase a bullet swage device. With this, if you lube the bullets, you can compress a .321 down to a .318 or .319 or whatever you determine to be best for your rifle. Not with solids of course, but with the typical run of 8mm bullets I have had good success.
You need to determine for yourself, but I find that dropping 10% off of the Minimum 8X57 Mauser loads, and loading up towards the low end of range, is not a bad range.
Best bet is to buy the S and B. Take along a chrony, and then load to that velocity with the same bullet weight. 196 grains is the standard for your time period, adn this is probably denoted on your flats by 14.4 grams, or something similar, which is teh metric equivalent for bullet weight. Only the marks on your rifle can tell you for sure.
Good luck. send me an email and I can share some loads I use with my Merkel prewar rifles for the same caliber. I dont do loads over the boards.