Those three are good but not the ultimate. I have built a lot of those and own all of them. The Lapua is a little faster and gets you closer to the 338-378. It also requires more expense for the big bolt face and brass. The 338-300 ultramag that some people on here call the edge in my opinion is the best choice of the three when all is compared. I did 20-30 of these back when remington first gave out the specs on the 300 ultra. Chris Dichter over at Pac-Nor, who I did a lot of business with and was in continuous phone contact, got the specs at the shot show I believe and within a couple weeks he had me about 15 barrels to try by necking up the 300 ultra to 338. By the time the 300's were hitting the shelf we were testing a bunch of the 338-300's. Results were that it is inherently accurate, gives good velocity, and can be built fairly inexpensively because it uses the standard mag bolt face. I had to modify some magazine boxes and actions slightly because of the length but nothing major. I had a thousand yard range outside the shop and never will forget the first one I did. I bore sighted, fired two shots at 200 yards to put it where I wanted, then shot a 3 5/8" group at 750 yards. First shots out of a new rifle. This cartridge is very good and is probably your most inexpensive option if you have a magnum action. Shawn on here has done a lot of these and probably has this one figured out pretty good by now.
Now, you said the ultimate. With all the research I have done to obtain the ultimate from a reasonable shoulder fired weapon considering currently available bullets for hunting I believe the chey-tac case modified and necked to 338 is the ultimate. Again a guy on here, Kirby Allen, does these. I have never talked to Kirby but I would imagine they would be expensive compared to the ultra's which are fairly easy to build. That action would cost you.
Next in line I would say is the 416 and 378 necked to 338. You can put these on a wby mk5 action which is easy to get. Open up the bolt face if it isn't the 378 action and get a 378 magazine box. These will get you a good 250+ fps over the ultramag line. Again the brass is expensive but to me it is worth it because these are my go to rifles when I get serious. I have seven 338-378's, six 338-300 ultra's and two 338 Lapuas. That should give you an opinion of what is the best from a guy who has built and shot a lot of all of them. I am not a long range shooter, but a mid range shooter. I typically can stalk to within a half mile of my target so between a quarter and half mile is where I do most of my shooting. My 338-378's will average 3550 -3750 fps with a 200 grain nosler ballistic tip. Inside a half mile the ballistics are incredible and the results are devastating on game. Outside a half mile go with the 300 matchking. The 200's kick like a 25-06, the 300's about like a 300 mag. That translates to accuracy for a lot of people. For the price the 416 or the 378 necked to 338 is the best option. If you have a few hundred more dollars the chey-tac necked to 338 I think is the ultimate.
The effect of a big 338 bullet on game at long range is devastating. It gives you a little room for error over the smaller calibers. In field conditions you may not always put that bullet exactly where you want. If you want to up the odds of your success use the 338 as minimum if you are a long range hunter. Shoot a lot at the range and see how many long range shots OFF THE BENCH would not be exactly through the chest. The big 338's will collect a lot of those animals. With smaller calibers you have a lot of wounding loss you don't want.
