Rifle cost shouldn't be your main concern when getting a .338LM - they are very expensive to feed for most folks used to shooting cheap ammo. Also, you're wasting time and money if you don't plan to shoot it beyond 1400 yards, otherwise a .300WM makes a lot more sense. But if you're still committed to .338LM, then definitely consider the Sako TRG-42. It's probably the best value out there for that cartridge. There are cheaper options out there, but I think with these precision rifles you definitely get what you pay for. The more expensive rifles like Ashbury and AI are excellent, but you might not see much of a difference over a mid-level rifle in the short term (with the exception of price). The switch-barrel rifles like the DTA SRS and the Barrett MRAD (I say "switch barrel" loosely with this rifle as 2 years on there are still no conversion kits available) are more expensive but you get the added feature of being able to have one platform with many rifles. That's only attractive to people who have a interest in having multiple precision rifles and for me it was a big deal because I bought one high-end scope that serves 4 rifle conversions. But again, the entry price is much higher.
So if you're new to this and especially if you don't have any experience making your own match ammo, seriously consider a .300WM first. TRGs are also made for that cartridge. And if you're set on .338LM, then make the commitment but consider the above advice when choosing a rifle.
ETA: the biggest mistake I see with people buying these rifles is they go out and spend several thousand on a quality rifle, then skimp on the optics. I can't tell you how many TRGs, AIs and MRADs I've seen with Bushnell, Redfield or Burris scopes (et al). These are magnum rifles and they will beat the shit out of cheap optics. I'm not saying you NEED to get a high-end scope, but why spend so much on a quality race car only to put cheap tires on it?