Truthfully the cheapest way to get a lighter rifle is to buy a new one as suggested earlier. To have Proof Research or Christensen arms rebarrel your Savage will cost more than that LW Hunter as suggested. I don't know if either will sell you a pre-fit barrel you can install yourself if you planned to DIY.
If you go with just the acton weight of a Savage you'll probably come in somewhere around 2.5 lbs so that will give you 4.5 lbs to play with to keep it under 7 lbs all up. A 22" fwt contour barrel will average around 1lb 13oz finishing .550" at the muzzle, a McMillan Hunters Edge will average 22 oz, Talley lwt mounts 4.5 oz, and Leupold 2.5-8X36 which is about the best variable scope for a lwt build will weigh in around 11.4 oz. So your roughly looking at a 6.75 lb rifle before you add a sling and ammo. You can trim about 3-4 oz off the build if you use a fixed power or 3-9X33 Ultralight by Leupold.
For the price to mold your Savage into a lightweight rifle you could buy a
Kimber Montana in .243 ($1200) or one of the new
Forbes rifles (not CLR, NULA or ULA) in .25-06 or .270 ($1300) for about the same money. The way I figure it a bare Savage action is worth $250-300, $350 for a prefit barrel, $500-700 for a lwt stock by the time it is finished, and $350-450 for scope and mounts. So for $1700 +/- if you were to buy all new parts you could come out with a rifle and be right at 7 lbs by the time you add a sling and ammo.
Sure you could do some other things and maybe shave another 4-6 oz off of the action by skeletonizing or running as a single shot. You'll probably carry the same amount of rounds hunting as you would if it had a magazine, so you might as well carry as many as you can in the rifle. Then to remove metal from the action and bolt would put your build over the $2K mark easily.