Back to the roots, that is, the original German specifications for barrels, bullets and loads!
Jon Speed in "Mauser -Original Oberndorf Sporting Rifles", pages 274-275, published the original Mauser factory specifications. According to these 1920s factory specs the 8x51K used the same barrel and bullet dimensions of the 8x57I, .318": barrel bore: 7.8 (tolerance + .05) mm = .307 - .309", groove 8.1 (+ .04, - .01) mm = .3188", twist one turn in 310 mm = 12.2".
Bullet diameter 8.09 (+ .04) mm = .3185 - .320", bullet weights 12.8 g = 197gr or 10.3g = 159 gr.
These 1920s Mauser specs are close to the current CIP ones: bore minimum 7.80 mm = .307", groove 8.07 mm = .3177" (Same as 8x57I, 8x60I, 8x64I), bullet maximum 8.07 mm = .3177".
The 1934 DWM handbook "Patrone und Schuss" advertized three loads, all using the same bullets as loaded in the 8x57I too:
Bullet number 117Y, 157 gr, in front of 2.5 g = 38.6 gr R5 powder for 657 m/s = 2156 fps, 20" test barrel (8x57I load using the same bullet: 3.0 g = 46 gr R5 for 837 m/s = 2746 fps, 26.8" test barrel)
Bullet # 299A, 12.7 g = 196 gr, 2.5 g = 38.6 gr R5 again, 640 m/s = 2100 fps ( 8x57I: 2.6 g Troisdorfer 1910 powder for 729 m/s = 2392 fps)
Bullet # 473A, 12.7 g = 196gr strong jacket soft nose, 2.50 g R5 again, 630 m/s = 2067 fps (8x57I: 2.75 g = 42.4 gr R5, 712 m/s = 2336 fps)
Note that the 8x51 K velocities were from a 51 cm = 20" barrel, while 8x57I ones are from a 68cm = 26.8" barrel.
So I would load the 8x51K with the same S&B 196gr .318" soft nose bullets I use in my 8x57I and reduce powder charges by 15 - 20% for a start.