I don't believe there is a single Winchester rifle I'd describe as a "sniper rifle" of any kind. Remington 700, or any Savage with an accutrigger is much closer to what you're looking for. 308win, is a good caliber, reasonably affordable, good availablity and flat shooting. I assume of course, you're looking for an accurate gun to hit paper (or other inanimate) targets at far distances. Of course the 6mm and 6.5mm calibers are some of the hot ones these days.
Look to spend about $700-800 on your rifle, and if you really want to get the best precision, at least the same on optics. Take the leftover from you $2000, and buy a reloading kit. You can load for yourself, way more accurate, and much cheaper ammo, than you could EVER buy for any price.
Actually Winchester did make a sniper rifle.
I'd look at a Remington 5R in .308 and a Sightron SIII 6.5-20x. That should put you very close to your budget limit. But... if you load that rifle with Federal Gold match 168 or 175 grain you'll have one heck of a nice shooting rig. This will hold you until you learn to reload.