The whole point of going to a larger cal is to push heavier bullets at elevated speeds to improve down range impact energy and to reduce drift.
So in the 338 world, the base is the Lapua and get's BIGGER... but with this much horsepower, cost and recoil also elevate in a big hurry.
If you want a 1000yds/m target rifle, look at the F Open world. Here the 7mm reigns with the 284 Win case the most popular. Recoil is moderate, bore life is very good, costs are not bad. Accuracy can be stunning...
Then there are plenty of 6.5's, 6mms and even the good old 308 as used in FTR that get out to 1000yds very well
Now if you want to put a hurt on something, bigger can be a whole lot of good.
For punching paper, I know the 7's are more accurate then the Lapua at 1000yds but somedays, you just want to make a big impression....so boils down to goals and priorities.
If you have your mind set on a 338cal something or other, the lapua is the easiest to set up if willing to get a action specific to the case. Otherwise, the RUM family can use reg magnum bolt heads... see 338 Edge or 338 RUM.
Anything smaller, may as well get a 300WM or 7RM/284
Jerry