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There are 600 yd. competitions held for AR's.  They are up to the task.  Did you expect another answer on AR15.com?  

Most manufacturers make rifles that are accurate enough to do this.  You will almost certainly want to handload to get the needed accuracy, though this is not absolutely required.  Your budget will allow the purchase of an excellent rifle, but you need about $500 more for good glass.  Good luck, and let us know what you decide.  

Quoted: can any of the AR-15s qualify as a "sniper"/"sharpshooter" weapon or is that reserved for the bolt actions exclusively?

Olympic Arms sells their top-of-the-line rifle called the "PCR-1" as a countersniper rifle and it is very accurate.  It features their Ultramatch barrel, which is generally acknowledged to be the best factory barrel available by any manufacturer.  You can also get them with the Denny Williams set trigger for about the price you mentioned--$1200.  The trigger of any rifle must be good for optimum accuracy and the William's set trigger is good for about 3# unset and 1.2-1.5# when set and absolutely crisp, breaking like glass, as they say.  They expect the rifle to shoot 1/4" groups at 100 yds., with handloads.  I recommend the 1:8.5" twist, if it is still available.  It will shoot the Sierra 69 gr. Matchkings very well, in most cases.  I have two and love them.  My brother has one, as well as four of my friends.  All of them are "shooters"--the rifles, as well as the men.  LOL  Good luck with whatever you do.  Please let us know, as we like to keep up with one another's new toys.  

www.olyarms.com