No, unless .006 difference in bullet diameter makes a difference and case head protrusion can be added, the 8mm56R is rimmed and does not have case head protrusion (.060 thousand rim), the 8mm57 has .110 thousands case head protrusion.
I would not attempt the project because of the straight pull bolt, I have a 303 Ross 1905, one of the most accurate rifles I have, the bolt operates like a M1 Garand without gas, it must not be a test but out of curiosity I stood the Ross on the but end with the barrel up and dropped a .300 down the barrel, the brass rod hit the bolt face, then the bolt flew open. I have no interest in pointing the rifle straight up and firing it, the bolt operates like it is gliding on glass, the rod could have bounced.
The 8mm56R is smaller than the 8mm57 in diameter, just another reason I believe it would not a good ideal.
F. Guffey