Go with the 338 Edge. Cheap good brass(Remington) and easy to find. Just neck up the 300 RUM, load and shoot. The 338 AM is an awsome cartridge and if you have ever dreamed of killing an animal at or beyond 1 MILE...that would be the cartridge to choose.
There are more than a dozen 338's, near bottom win, then 340 WBY, Next, these are close 338 Norma,RUM,Edge,Lapua,step up LaIMP, Kahn, ultra max,AX, THEN 338-378,Raptor,AM.And I know I missed some Gibs,Sin?
The Lazzeroni and the 338-378 wby are big heavy hitters that put you into the big 338 category and the ones to look at if that is what you want. The 338-378 wby has been about as good as it gets with a big 338 since several started wildcatting it in the 70's. After 40 years it is still at the top in a hunting weight rifle. I have seven of them. If you want a heavy hitter it is an excellent choice.
The edge mentioned on here is one of the many names given to the 338-300 RUM. It and the 338 RUM which is a factory cartridge and has the same performance are on the low end beating the 340 wby by about 125 fps. Those two are excellent choices for a guy wanting to get started cheaply with a donor magnum action and learn the ropes of long range before getting into the big 338's. I did quite a few 338-300 RUM's for guys from 1998-2001 because they were the biggest thing out there that could be done cheaply on a standard magnum action and got a little more performance than the 340 wby which was the best until then. In 2001 the 338 RUM came out with the same performance as the 338-300 RUM and fit better in the rem 700 action so was the better choice from then on. The 338-378 wby and similar size cartridges will outperform the RUM's by 225-300 fps depending on the rifle. It outperforms the 338 lapua by about 200 fps. The Lazzeroni is between these and the 338-378. The Excalibur case improved also gives top performance.
The problem with going to a big 338 is cost. So that is the decision. How much performance do you want to pay for.
Predictions are difficult, especially when they involve the future