Ricknolan, What some call the edge on here is simply the 300 ultramag necked to 338. Pac-Nor will be glad to send you a barrel to screw into your Savage chambered in it. It has been a very common chambering over the past ten years used by quite a few gunsmiths. It has a small velocity gain over the standard 338 ultramag which is just a tad shorter. Both require a muzzle brake so include that into your cost.
At the range you mentioned inside 700 yards shooting the 338 winchester is a great option and most shoot great right off the shelf with a little accurizing. I have two and they both shoot great. Again it is usually just a very accurate round in a good rifle with proper loads. I used one in Alaska about ten years ago as a back up rifle. I took a 67" moose at nearly 1100 yards and two big bull caribou at 500 and 700. Very good performance on all animals. The 338 win mag is at it's best with the 200 grain nosler ballistic tips at 3000 fps. That shortenned magnum case on the winny doesn't work well with heavy, long bullets and it just doesn't get the velocity. You do not need a muzzle brake with this one shooting the 200 grain bullet. It is quite pleasant with that load. It shoots flat enough and hits plenty hard enough for anything at your range. Cheap, over the counter and let a good gunsmith accirize it for you. I think you will be very satisfied on a budget. Then take the money you saved and go on a good elk hunt.
If you want to go big time then get a gunsmith to screw in a 338-300 ultramag barrel for you and put a brake on it. It is a very accurate round and will throw a 300 grain matchking a long way accurately.
J E Custom, Here are some velocities you can expect from the rounds you mentioned with 250 grain bullets. I have built and shot all of those for years and have volumous data.
340 wby over 3000 fps
338 Lapua over 3150 fps
338-300 ultramag over 3100 fps
338-378 wby over 3250 fps
These are averages. Some guns may not get that and some may get a lot more. Just depends on the barrel. I have three 7mm stw's that will not shoot a bit faster than my 7mm wby's, two will shoot faster, but none as fast as my 7mm-300 wby. The stw has more case capacity though. I have gone through who knows how many 300 wby's through the years but kept the best one I have ever seen. The velocities out of it equal the 30-378 wby and it only has a 26" tube. 200 grain bullet at 3250, 180 at 3475. I could go on and on. So I always just talk about averages because somebody is always going to have a gun that just blows the average out of the water.
Predictions are difficult, especially when they involve the future