My next question, jrock!
But, without waiting for an answer, I'll give a few choices and the reasoning behind them.
First, you want something light, bolt action, synthetic, and maybe stainless. To me, that would point to something from a short action, with a 20-24" sporter weight barrel.
Second, you want a faster cartridge, but plan on not shooting terribly far. That is something we can work with! Honestly, though, for what you are using it for, something along the lines of .243, .260, 7-08, or 308 would work well, but I'll stick with the faster cartridges. Since I am working in the frame of a short action, I would use a WSM since they are relatively fast in a factory loading.
My first suggestion would be another Savage-hell, they are priced well, and accurate to boot. Savage makes several models that would fit the bill quite nicely. People complain of the stocks, but even with the factory stocks, I've not had problems. A better one could always be swapped in.
The Weather Warrior (16 series) is light (6.5-6.75 lb), stainless, and available in the WSMs. It is available in the 16FSS (blind mag), 16FHSS (hinged floorplate), or 16FCSS (box mag).
The Hunters (11 series) are availble in the same calibers and configurations, but blued.
Stevens would also be a consideration (you don't get the nice Accutrigger, however). But, the WSMs are not available, and you have to go with the long action magnums. Price is great on them, though.
Next, Remington has several choices. First, as Blackduck said, the Model 7. A .5" shorter action than the 700 along with shorter barrels make it very light and handy to carry.
The XCR Camo is synthetic, stainless, and available in WSMs and RSUMs. I do wonder about the life of the RSUMs, as I sure don't see them on the shelves of the local gunshops much. It is a touch over 7lbs.
The 700 sure isn't a bad choice either, just a bit heavier and longer.
The XCR is synthetic, trinyte (corrosion coated), available in WSM, and is 7 3/8 lb. A little heavier, but it also helps with felt recoil.
The SPS is available in blued or stainless, a bit cheaper than the other versions and configured like the XCR. Like the Savage, people have complained of the cheap stock, but swap one in later if you wish.
Or, pick up a used M70 featherweight in WSM. They are a bit hard to find in synthetic, and I really can't remember if they made a synthetic WSM. A synthetic could always be swapped in later if you can't find one already with one.