My recommendation would be a light weight .308 Win. It's a great light mountain rifle. As I prefer the CRF (and can't imagine that there's not a Winchester classic fwt in LH in the USA) and get a light weight synthetic stock for it. I'd leave the barrel at 22" not much difference from 20" and still a nice, handy, fast-swinging rifle.
I have a pre 64 M70 fwt in .308 Win (22" barrel). It's a real performer. 150 gr bullets @ about 2,950 fps makes it a real hard hitter out to 300 yards. Might get a synthetic stock for it for hunting (though the pre 64 die hards may disapprove) it will keep the original stock looking nice and give me a real light weight mountain rifle. There's a well know hunting writer here in Oz (that's Australia) whose favourite deer rifle for many years was a very light weight Remington 600 with a synthetic stock and a soda straw thin barrel (all up weight with scope was about 5.5 lbs).
I think the .308 Win is a very under estimated cartridge. Whilst the 7MM-08 is a great round (because it is based on the great little .308 Win) the .308 Win with good 150gr bullets will have the edge on the 7MM with comparable (SD and BC) bullets (i.e. of around 139gr weight). If you use bullets any heavier than that in the 7MM-08 it's velocity (and tradjectory) is inferior to that of the .308 Win with 150gr bullets loaded to say 2,950fps for all reasonable hunting distances. But the .308 hits harder (bigger bullet diameter and heavier weight).
Ran them both through "Point Blank" and they are so close together it is canny. Though the .308 has 200 ft lbs more energy at 300 yards than the 7-08. Other ballistics, drop, retained velocity etc, are fairly similar.
The legendary advantages of the 7MM (long, high BC) bullets are only realised in the heavier match bullets (160gr plus) and when pushed at 7MM Rem Mag velocities. Hence the immense popularity of this round.
But for a light short actioned mountain rifle for deer you can't beat a .308 Win, though the 7-08 comes close to equalling it, but it doesn't better it. Plus the .308 Win legendary accuracy. My pre 64 M70 fwt is a real tack driver and (we) frequetly shoot .5" 3 shot groups @ 100 yards. Gives you great confidence in it as a hunting rifle.
Magnum308