While I'm also of the opinion that '06 or .270 would be fine, just taking a quick look at cdnn, I see a bunch of choices in 300 win mag, here are a few:
Weatherby Mark 5 synthetic in 300, or 7mm mag - $799
Weatherby Vanguard deluxe walnut in 300 - $599
New model (with pre '64 bolt design) Winchester 70 sporter magnum in 300 or 7mm mag - $679
New model Winchester 70 featherweight magnum in 300 - $649
Of course, a Tikka t3 or if you want to go up scale Sako 85 would fit your bill just as well, as would various Remingtons and Savages. Add on a Zeiss conquest 3-9 for $400 and any of the rifles above (except the Sako) should come in well below your minimum price.... I don't really see a reason to drop $2500 on this rifle unless you're looking for pretty walnut, engraving, etc.
As an aside, I have a coworker who is stationed in Southern NM who has taken his fair share of Elk and Oryx. When I asked him what he uses to hunt big game, he said he has started using 7mm mag across the board. This comes from a man who has an impressive arsenal of big game and dangerous game rifles, he just doesn't think anything more is needed for animals in the Elk/Oryx class. Just some food for thought.