Clearly any powerful rifle chambering intended for use in a target rifle must be moderated by some sort of recoil dampening devise, most often a muzzle brake, and heavy gun weight. It should be easier to shoot long strings with a .338 Winchester than with a .338 Lapua, but I don't know if the ballistic advantage of the Lapua is that far ahead. The Lapua drives a 250 gr MK at 2950 fps, and a .338 Winchester will drive that same bullet at 2700. The Lapua bullet goes subsonic at about 1450 yards at sea level, and the Winchester drops subsonic at about 1290. Depending on several factors, including the amount of time spent transonic and the amount of spin on the bullet, the bullet may or may not loose stability as it passes through the transonic velocity, but if the bullet fired from a Lapua chambered rifle becomes unstable while the bullet from the Winchester cartridge does not, the practical range would be much closer. Of course the opposite could happen, where the bullet from the Lapua rifle remained stable and the Winchester did not, in which case the Lapua would have a significant advantage.
If I was contemplating either of these cartridges for long range target shooting it would be in a custom rifle with a barrel as long and heavy as I could arrange. I would choose the .338 Winchester simply because the cost of the Lapua brass is prohibitive, and the Winchester can produce similar ballistics with less powder. It is sort of a .308-.30/06 comparison, but like the .30/06, the .338 Lapua walks away from the .338 Winchester if you choose heavy bullets like the 300 gr MK. All things being equal, due to its larger bore size, the .338 Winchester can fire the same weight of bullet weight faster than the .300 can, although the smaller bullet will have a better BC, so it is unclear if the .300 with a 240 gr MK would be any worse than a 250 gr .338. The .338 Winchester like the .300 Winchester suffers from a short neck, so I would have the chamber cut long so the bullet could be seated with only the boat-tail extending into the shoulder. This would help to reduce the velocity advantage attributed to the Lapua round by giving the .338 Winchester just a bit more.