You absolutely can't go wrong with a .300WM for elk and moose, as well as deer. 300RUMs are great, but a 300WM will do it all the same. The 300WSM doesn't offer any more power over the 300WM, it only offers a weight savings since its a short action. The 338Lapua is way more than necessary and you're going to send about $5 down range every time you pull the trigger. Ammo isn't cheap for those. Here is a comment I put in a recent inquiry about an "all-around western rifle". I have a 300WM and a 30-06. Both are great elk guns and can take an elk out to 500. The 300WM offers better performance beyond that. As for deer running off and dying, that's more the norm than those "DRT" shows. I've even helped pack out elk that were killed with a .243. If you're ever worried about recoil, spend the money on a muzzle brake and never look back. Just carry a lightweight set of hearing protection.
From a previous post:
"Several great options have been mentioned above. You can't go wrong with 300wm, 30-06, or 7mmRM. Bullet choice should definitely be a factor and all have great bullet choices for all species (165gr-190gr). If you're going to hand load, then ammo availability isn't a factor. There are also other great choices, including the 270, 270wsm, 300wsm, 280AI (working on a build), and 28 Nosler. You can go to a western hunting forum, ask this question, and get 25 different answers. In any case, I recommend a lighter trigger and a muzzle brake for whatever you choose. A good set of lightweight ear plugs can easily mitigate the increased noise of a brake. All of my new rifles get a brake, regardless of what they're chambered for.
To the 6.5 CM: it will work perfectly on elk to beyond 200 yards. Plenty of elk have dropped from one shot from a 6.5CM from 400+ yards, but those are hunters who are practicing regularly. As with any hunting situation, practice is key, especially at ranges you think you'll be shooting. My one and only beef with the 6.5CM (and why I'll never own one) is that it is a bandwagon cartridge. Don't get me wrong, its a great cartridge, but so many folks out here have them (we call them the Creedmoor Army) and they are very in your face about how its the best cartridge ever and you're stupid if you don't have one. Admittedly, my anti-establishment personality plays into that. No, I won't jump off a bridge if all my friends are doing it and that's how I see the 6.5CM.
Elk are extremely tough and getting them to drop in their tracks is not the norm. My daughter had to shoot her bull three times to get it to fall and all three shots were on the money. bwese is right. If its still standing, shoot until its not standing. My daughter shoots a 7mm-08, which is a great short-range (400 yds and under) elk rifle. As for brands, I wouldn't hesitate to look at Savage. They have surpassed most other for out-of-the-box accuracy for a budget-minded rifle. I was a diehard Remington guy until it was time to buy my kids their rifles. At the same time, Tikka, Bergara, and Winchester are all hard to beat. Remington is finally coming back around to the quality 700s they were known for up until about 10 years ago."