It seems to be easier to purchase an ultralight hunting rifle than it is to have one built, but that has not deterred me.
If we start with the action, a push feed should be lighter than a CRF action if both are the same length and made from the same materials, although Kimber seems to be able to defy this logic. For a custom build, my choice is a Remington 700. I like 700s and there's more after market stuff out there for 700s and their clones than there is for any other action.
There are some pretty light triggers around, but there are some very heavy ones as well. For a project like this I'd pass on a Jewell, or a Huber Concepts trigger in preference for the aluminum Calvin Elite Timney. We can reduce striker weight by going to a titanium firing pin, and we can opt for an aluminum bolt shroud, and the bolt body can be fluted. The bolt handle can be skeletonized. An aluminum follower can replace the steel follower, and the magazine box could be made of plastic or aluminum rather than steel, and bottom metal could be plastic or aluminum. A blind magazine might be slightly lighter than a floor-plate, I'd choose the floor-plate anyway, and a floor-plate is definitely lighter than a DBM, even Remington's crappy DBM.
The barrel contour should be the lightest deemed safe for caliber, and if not the lightest possible, it can be fluted for further weight reduction. While its easy to say that wacking off a few inches of barrel is the simplest method of reducing weight, it can also make the rifle clubby, slow to get into action, and difficult to hold on target. For that reason, the light contour barrel should be long enough to balance the rifle slightly towards the muzzle. For my switch barrel, I chose a Ron Smith stainless 22" #3 contour which is as light as is reasonable for a .375 bore. The other barrels have smaller bores (6mm & 7mm) and their length of 24" was chosen for performance over weight without becoming too muzzle heavy.
I like McMillan stocks, but with their long delivery time, instead chose the Ultralight stock made by Wildcat Composites in Sherwood Park AB. With a 13.5" LOP the stock weighs 17 oz, sans recoil pad. My rifle is coming in a bit heavier than the Kimber, but the Kimber doesn't come in .375 Scovile either.