If you want to "out Shoot" a 338 Lapua at 2000 yards. Other than shooter skill the only way the rifle will make it easier is with a higher BC bullet at the same or more velocity. Assuming they are shooting the 300 Berger then you will need to do it with more velocity. That means you will need a bigger case such as the 338 Lapua AI , 338 Snipe Tac , or one of the variations off the 338-378 Wby Case. Or Talk to Defensive Edge about a 338 EDGE +P or 338 Terminator +P . Or, you could venture into the world of the 375's, or larger.
But looking for a "smaller, less expensive," caliber to out shoot a 338 Lapua , it ain't going to happen unless you live in a world of make believe. At 2000 yards BC and retained velocity along with less drift will rule as far as caliber capabilities go. But again shooter skill will be a big plus with any rifle.
Jeff