Savage heavy barrel.....AccuTrigger....caliber?..308 if you don't currently reload, 260, 6.5-284Norma, .300WSM or WinMag if you do reload...I would throw in the 338Lapua but I don't think there's any "budget" associated with this round.
Nikon is the only scope that comes to mind when thinking budget and still perform out to 1K....I used a BuckMaster 6-18x40 on my .300WinMag and it performed flawlessly in this area. This scope runs under $400 but you get what you pay for with optics.
Makes no sense to buy a new rifle and tear it up.....buy the Savage...chances are it'll shoot better than most men or machines....if you decide to swap the barrel, YOU can do it and not have to pay someone else. The AccuTrigger is more than sufficient for 1K shooting, user friendly and adjustable. I will also add that the factory barrels on Savage rifles, IMO, are very good. The last one I bought needed no break-in whatsoever. The first 100yd, 3-shot group measured .346" and the last 3-shot group at 1500yds, two weeks ago measured 10".
All the "other" rifles I have owned needed more rounds down the barrel, with lots of cleaning, for barrel break-in and the triggers had to be worked over, adjusted a lot or replaced.
"Purty is as purty does". My money's on Savage. JohnnyK.