I think the .316dia is a bit on the small side for use in early 8mm Mauser cal rifles with the 'J' bore,though could probably be made to fly nicely with the right load. Should be a thou or 2 over the bore size to start.
It does indeed sound like a bullet for the 8.15x46R. A little heavy for the 310 Cadet,,then again maybe a bit light weight for the 8.15.
The gas check is odd for either application. But American shooters like G/C lead bullets..they can go faster!..
The 8x56R is .330 bullet dia. The earlier version that it's based on,, the 8x50R, is .323dia.
Most all of the Austrian M95's straight pull rifles & carbines were redone to the 8x56R from their original 8x50R caliber in the 1930's. Most rifles were cut down to carbine length in the conversion process.
The Lee-Speed Sporter Bolt rifles were available in '8mm Mannlicher' which was the 8x50R caliber.
I just sold my M95 rifle and M95/34 carbine and the last 1000rds of ammo for them. All 1937 & '38 mfg ammo in 8x56R. Quite a shoulder bruiser in the carbine.,,and don't creep up on that stock!!