This may not be a problem for others, but I won't ever own another hunting rifle without either a 3 position or tang safety. I had a Colt Light Rifle (cheaper factory version of Forbes design made for a while) that shot and packed great. I really loved that gun but the 2 position safety was always rubbing off into the fire position on my pack when I carried the rifle slung over my left shoulder. I like to switch shoulders to carry the gun on a lot while hunting depending on terrain and fatigue. I had my gunsmith drill out the hole for the detent ball to go into a little bigger and put a stronger spring in but the safety still did it. Even without carrying a round in the chamber it allowed my bolt to open and exposed my action to debris. I ended up selling the rifle.
I have two custom M70's for my big game hunting, a classic stainless in 264 win mag that is my favorite rifle and has been a great interior decorator putting critters on my wall. Only downside of it is that it weighs 10.5lbs with glass, sling, and ammo, ready to hunt. Good for the flat ground and high winds of the plains, but a little much for steep country. For mountains I'm having Dad's old 270 rebuilt with a B&C lightweight aluminum bedded stock, #3 fluted Pac Nor barrel, and Pacific Tool and Gauge aluminum bottom metal. Weighing all the components my ready to hunt weight should be 8.3lbs. Still not ultralight, but I like heavier higher magnification glass that adds weight. I'm hoping to have it back any day now and have a box of 150gr accubond long range bullets ready to load.
Factory guns I'd consider would be the Browing A bolt or X bolt, I've owned two browning A bolts that shot really well. The A bolt SS in 300WSM with a B&C Alaskan stock, Talley rings, a trigger kit, and Vias removable muzzle break would fit in your budget with room left for a mid-range Vortex or Leupold scope. Savage rifles are great also but I don't think they make anything as light as the Browning in a magnum. Ruger is the same way I think. My research is a year or two old so I might be wrong. I have a Ruger American in 270 that is extremely accurate also, I don't think they can be beat for the money. The first handload I tried averages under .7" for 3 shot groups at 100yds, but the barrel heats up quick and the 4th and 5th begin to open the group. The Ruger also has a pretty flimsy stock that can contact the barrel using a sling or bipod and putting on a fair amount of pressure. I wish B&C or someone would make an aftermarket stiffer stock for this rifle, it performs above it's price.
Another option I've considered is building a Remington 700 and adding the PT&G 3 position safety. If you want to customize one for you anything made for a rifle is made for the 700.