7mm08 or .280 or .308?

[Cossack]

I shoot 2 out of 3 now. The 280 is my idea of a 'deer rifle' that can be 'stretched' both for distance and weight of bullet. Mine is a Remi Mt Rifle, loaded with 150 Barnes X it will do justice to your elk rifle need . And with 120's would be about right for antilope. Weighs 7.8 lbs trailside, with 2-7 Leupold VXII and 5. Been used on deer, elk and (would you believe) Ibex. Accurate, handy and versatile.
The 7/08 is a pure deer rifle. It's a Kimber Montana and weighs in at 6.5 trailside, with 1.75-6 Leupold VXIII and 5. A pure pleasure to shoot (I too don't like recoil) and to carry. I load it with 140 Barnes TSX for 2950 fps. It's new so only one kill but that was very telling. Only problem is that it's a short action and smaller case so bullet weight options are limited.
Shot a 308 for a while too, unremarkable.

Bottom line of 3 you specify=280.