Remington offers their model 700 in a police sniper model, in your price range, but what kind of accuracy are you looking fore? It will give you a hunting type of accuracy, but definitely not sniper quality.
All "sniper" rifles are highly accurized hunting rifles. They are torn apart, modified, and rebuilt. The trigger is replaced, the barrel is replaced, and sometimes even the stock. The only thing original is the action, and even that has been re machined to give tighter tolerances in bolt lockup and overall "squareness", meaning the receiver face and bolt lugs are re machined to be perfectly square to the bolt and bolt face.
Also, if you are spending $1,000 on a bare bones rifle, be prepared to spend at least $500 on a good scope.
Leupold vari-x III, Burris Black diamond, Nightforce, etc. are higher dollar higher quality scopes.
But in a nutshell, you are asking for 3 separate rifles. One for hunting, one for competition, and one for sheer oh my god that is the most awesome looking rifle I got to have one too!!
A Springfield M1A can do all three, but the price tag starts at about 1,200 - 1,400 dollars.