The Lipsey's website (distributor) is a good way to find out what factory rifles are out there chambered in a specific cartridge at the moment:
7mm-08
280 Ackley Improved
7mm PRC
7mm Rem Mag
If you're going ultralight, keep in mind that recoil will be more of a factor earlier on than with heavier rifles, so a 7PRC will have more felt recoil in a 6lb gun than a 10lb gun. What other people may be comfortable with in an average or heavier gun may simply be unpleasant in a lightweight mountain gun. Pretty sure I've listed those cartridges in order of least to most recoil. Keep in mind though that as a general rule, the lighter you want a gun, the more expensive it gets, and the point of diminishing returns kicks in pretty quickly. Whatever you do though, don't get hung up on ballistics - any of these are good for western big game at ethical ranges. Focus on selecting a good big-game bullet, and practicing from field-realistic positions, and you'll be good to go.
As to what gun specifically, not too many people will outshoot a Howa, Savage, or a Ruger American, and those are on the cheap end. Bergara's are known to be excellent shooters for their price point as well. But a
Tikka T3X Light in 7mm-08 would be a great gun out here, especially if you want light, accurate, manageable recoil, and a .284 bullet, all for under a $1000.