If you are set on a Weatherby, I personally think it begs for a Wby cartridge. With Mule Deer and Elk on your list....I would make 7mm a minimum size...and think the 300 Weatherby would be choice...as lots of choices in bullet weights, bullet styles, etc....as versatile as the 30-06 or 300 wm...but with heavier bullets can rival a 338 for those longer rutting bull elk or grizzly needs....or even moose no matter if Yukon, Shiras, or Newfoundland hunted.
If I was going for a non-Weatherby cartridge it would hands down be 280 AI.... I just became aware of them a few years ago and bought my first last year in a Browning XBolt Pro Carbon and absolutely love it. It is what I grab most of the time last year and will be this year....even above my fav overall rifle that I have hunted with since 90s...ABolt 30-06...or a newer Begara B14 Squared Crest in 6.5 PRC.
That 280 AI is just a special round.....
When contemplating an elk, caribou, or moose rifle....that is suitable for large grizzlies as well in case my shot is their dinner bell....I am torn between 300 Wby or 340 Wby for Weatherby family......or a nice 35 Whelen to really shake things up. LOL