I realize that the question was "How Light" and it needs to be thought about before entering into a build like this In my opinion.
The only advantage to a light weight rifle is Weight. All other advantages go to the heavier rifle.
In a hunting situation you normally don't have a good steady rest and the light weight rifle has a bad habit of moving around and can be effected by even your heart beat.
Recoil is also a consideration unless it will be a very small cartridge, and most on this site need a much larger caliber for distance. I built a 338 RCM on a Mod 7 rem that weighted just below 6 pounds without the scope and found it to be impossible to shoot a follow up shot on running game due to muzzle rise and recoil. so I had to add a muzzle brake that weighted just shy of 1 pound. so with the added weight of the scope and muzzle brake I am now approaching 7.5 pounds. (Kind of defeated the whole reason to build a ultra light rifle).
The next downside to a ultra light rifle is the difficulty in finding an accurate load. Light weight barrels are finicky and require more time to cool between shots as mentioned.
The real puzzle to me is why go to a light rifle when a "Heavy" rifle only weighs 2 or 3 pounds more. I hunted Colorado with a 13 Pound 338 and was glad i did, especially when I was winded from climbing at High altitude and needed a steady hold for long shots.
The little 338 RCM is great and very handy, but there are two many advantages to a medium to heavy rifle to worry about 2 or 3 pounds. If you plan on hunting close where absolute accuracy Is not needed it is nice and my little 338 Is devastating on hogs.
Just the other side of building an ultra light rifle and something to think about before spending the money.
Just My opinion
J E CUSTOM