Robert - If you will handload, don't count out the 8x57mm Mauser. Lots of cheap factory ammo for practice, and 200gr or 220gr bullets should knock the tar out of what you're after at those ranges.

Of the ones you mention, I like the .338-06. Hits like a ton of bricks.

Also, I really wouldn't be surprised if Winchester came out with a .338 WSM real soon. Medium magnum performance in a short case may be exactly what you're after.

Good Dogwork and Good Hunting