Unlike most folks,
I size up your list, identify the biggestand toughest critter and recommendfor that onerather than the more common "go middle of the road" and then hope with a little luck it will take care of the toughest game.

The rest of the critters on your listare pretty puny and weak compared to elk and that kind of distance weakens all cartridges substantially, IMHO you need a fairly salty round if you are even going to get close to the concept of b-e-s-t.

On the low end, I'd look at a 300 RUM, Weatherby, or Lazzeroniwhile on the upper end, a 338 Lapua or 338-378 Weatherby. 338 RUM and 340 Weatherby in the middle. Probably handload with Barnes MRX bullets, scope with tactical turrets with externally readable MOA adjustments possibly with a Piccany (sp?) slope mount, range finder, pre-printed ballistics table. For the rifle I'd use a 12 to 13 pound rifle (scoped, loaded, bi-podded and slung) with custom fitted synthetic or laminated stock with all the trick bedding,thick HD recoil pad, mercury recoil reducer if the weight wasn't coming up properlyand probably utilize a VAIS muzzlebrake. Preferrably a two man sniper team approach with Walker Game Ears for sure.

The best rifle for 600 yard elk shooting isn't going to be the same as the mountain rifle you're going to creep thru the dark timber with nor one you'll find at Walmart. You'll want to practice and be proficeint at 800 to 1000 yards.... then 600 yardswill seem easy. Some long range training with the pros would be advisable.

Specialized job request, specialized tools as a result.... not too hard.

Of course, someone out there is going to want to use a 220 Swift and do it all at night.