I would recommend a rifle similiar to the one in this pic:
Chambering is not overly important really as there are dozens of chambering that will work very well. I would recommend at least a mid sized 7mm cal magnum as a minimum up to the big 338 mags if you want that much recoil. Just remember in a lightweight rifle, the big 338 magnums will have significantly more recoil then the smaller calibers although muuzzle brakes tame recoil significantly.
The rifle in the big is a Rem 700 long action, same as the one you have bedded in an HS Sporter stock and fitted with an HS Detachable magazine system. The barrel is a light weight #3 contour 27" Lilja fitted with a Defensive Edge Muzzle brake.
To keep weight down I used a set of two piece Burris double dovetail bases with a set of burris signature DD rings.
The scope is a Leupold Mk4 3.5-10x 40mm First Focal Plane with the TMR reticle. All metal is coated by Shawn Carlock in flat black teflon.
With the Harris bipod the rifle weighs in the 9.5 lb range including sling ready to hunt with. It is easily a 1/2 mile rifle, 1000 yards is pushing it a bit with this particular rifle but it would certainly do it on elk size game.
You could get any round in a rifle like this from a 7mm STW up to a 338 Edge or Lapua if you wanted as well. Even a 375 RUM would fit well into a rifle like this.
With the super bullets out today there is no requirement for a 338 cal magnum for elk but they do offer more authority on impact then the smaller calibers.
The rifle in the pic will be with me on my guided elk hunt next fall. It is chambered in my 7mm AM and loaded with the 200 gr ULD RBBT to 3150 fps I have no worries at all pointing it at a 1000 lb bull elk at any range out to 800 yards.
You can go lighter then this but then you really start compromising on consistancy at long range so it depends on what you plan to do with the rifle.