I think you'll have fun with a 375-08. I'm just about finished with a VZ-24 that I chambered for the .375 Whelen Improved which I imagine is close to the 375-08. I shot it last week for the first time to check sight regulation using fireformed cases, 55grs of H4895 and 235 gr Speer bullet which choronographed at 2525 fps. With absolutely no sign of excessive pressures, I'm confident this chambering will get 2600 fps with 250 grain bullets.

30-06 brass isn't any harder to find than .308 and the cartidge feeds beautifully without any rail alterations.

Good luck with it.