First off, welcome to a great place and a great way to hunt. There are lots of useful articles on the home page if you haven't visited.
For LR hunting, you must first get lots of practise. The 300RUM and 338 Edge are going to be expensive and tiring to practise with. Not to mention hell on your barrel.
How far do you intend to hunt? If you haven't spent time shooting LR, it is quite the learning curve so having a super duper rifle doesn't mean you can put the bullet where it matters. Only practise will do that.
So if you are looking at 700yds and in, why not the 300WM, 338WM, or 300Wby. All are plenty powerful with the 300WM the most common, so easy to set up.
My suggestion would be to look at a Savage heavy barrel in 300WM. They have some very nice laminated stock versions. Put a good brake on it and you are making noise for around $800. The accutrigger is a superb trigger so you will not need to spend money to swap that. The savage rifles once bedded will shoot 1/2 MOA in general. That is plenty accurate for most any LR hunting application and equal to many custom hunting rifles
Spend the extra money on ammo, scope and time at the range/field. Expect to burn out that barrel in 1500rds so money for a replacement is in order. If you practised once a week and shot 50rds each outing, that barrel will be toast by next summer. The RUM and edge would be toast alot sooner.
Cartridges like the RUM and Edge really shine when you are pushing very heavy for calibre bullets in very long barrels. The difference between a 300WM and a RUM in a 26" pipe and 208gr Amax is about 100fps. For this slight increase in velocity you are burning 40% more powder and 2/3 the barrel life. Then there is significantly more noise/recoil.
No elk is ever going to complain about being poked by the slower rd and will die just as quickly.
PS if you really want to do some practise, with the extra money you now have, pick up a Savage HB in 223. Now you can practise for pennies and your hunting rifle will last a few seasons. Just shoot the 75gr Amax at 3/4 the distance of your 300WM and you will get the same doping.