What's your definition of "lightweight mountain rifle"? I can tell you that what I consider to be a lightweight mountain rifle doesn't really go with a .338 caliber cartridge with 1000 yd capabilities, that combo would be almost unshootable to me because of recoil. You could brake it but that requires hearing protection while hunting which I personally don't want to use. The only barrel maker on that list that's going to make you a stainless barrel in .338 in any contour small enough to keep the weight reasonable is Hart, maybe Benchmark too but I'm not sure if they'll build a lightweight stainless barrel. You could use a chrome moly barrel but I only use stainless, no chrome moly for me. I'm just saying that what you're proposing to build doesn't seem very practical. Everybody says they want a 1000 yd. hunting rifle that weighs 6 lbs. but in reality they don't work very well. Rifles capable of consistent 1000 yd. hits on game are by necessity fairly heavy.
If it were me, I'd focus on something in the 6.5mm or 7mm range. There are plenty of great high B.C. bullets available for that. It'd use the new Defiance machine controlled round feed action, a Jewell trigger, a Mcmillian stock in your choice of patterns with Edge technology, and a Bartlein or Krieger barrel in a contour light enough to make it balance correctly. The barrel would be no longer than 26" and it would not have a brake.
If you want a 1000 yd. 338 then build one, but to build it where it'll be shootable it's going to have to have some mass to it, otherwise you'll just end up with an uncontrollable monster.