The 7mm Rem Mag is a great long range cartridge, and would be a much more reasonable choice than the 338LM. I've got a Lapua Mag, but it's heavy, and ammo is not cheap (even reloading components set you back over a dollar a shot).
Even a good .308 would serve well for long range shooting.
But the problem is, most folks somehow think that just owning the rifle and a cool "ooh-ahh" scope is all they need. We continue to be amazed at our long range clinics and even our long range shooting matches just how ignorant many folks are of the fundamentals of hitting long range targets. We see lots of deflated egos, no kidding...
Without proper training, it's next to impossible to make a shot even at 500 yards--let alone 1000 yards. The only way to learn is to have someone to teach you. The equipment is really the easy part. The cartridge choice is not even so important (there are many good choices)... it's the knowledge that is incredibly lacking...
We recently held a long range match where a fellow scored 65 points out of a possible 777 points on plates ranging from 350 yards to 1050 yards... he had a custom short mag rifle, and a Swarovski scope, and had no clue as to what to do with it.
Dan