Well, the 338 Lapua will do all you need and more. It is a very capable caliber to 1K+ with 300grn bullets and farther. 1500yds would be about max in my opinion for elk. That would be the pros. The cons, they are expensive to feed, and they are generally a heavy rifle. It does have a lot of recoil unless you have a good muzzle break. You can make a lighter version, but you will have to buy some very specific things to do that. The actions for a 338 Lapua are heavier and longer than a normal action.
So with that said, here's a couple of ideas. My cousin currently owns a Lapua. It is build off of the Savage Target action. You will only be able to single feed the rounds. It will eject a fired case, but you will have to pull the bolt to eject a loaded round. IMO, this isn't much of a concern when long range hunting. You would be able to put this on a lighter stock. He currently is using a 32" Douglas heavy barrel, but you could probably go down to around a 26-28" with a varmint contour. I would suggest buying a Muscle Break, APS Painkiller, or a Fat Bastard. These all seem to be really good breaks for heavy calibers. You will need to make sure the stock you get if you should choose the target action has the 3 holes for the action screws. This is if you want to build your own.
So far the feedback on the Savage 111LRH Lapua, and 110FCP-k with the H-S stock are getting some really good reviews. As for other models, if you can come up with the cash you have the Sako TRG42, and Accuracy International AWS. These latter rifles are really pricey, but worth it if you can afford them.
For scopes, you should consider the Vortex Viper PST, Viper HS, Razor HD, SWFA 5-20 HD, Bushnell Elite 4200 Tactical, Leupy MK4, Night Force 5.5-22 or some variant there in, or the Sightron SIII series power of your choice. Heck, I'm going to throw in the Weaver Tactical 5-15, and 4-20. They look to be very solidly built.
Sooooo with all of that said, how about a 338 RUM? It is cheaper to load for, and really won't lose out to the Lapua that much. 300 RUM with a the Berger 210-230, 240SMK will also do it. I hope you are going to be reloading because that is the only way you will be able to get either of these calibers to work reliably for you to 1K. What's your budget? How ofter are you going to use this rifle? Is it a one time deal? Just some things to think about.
Tank