For a coyote rifle, get something that packs more punch than a .223/5.56mm. Even though coyotes are pests, it's not right to injure them.
And don't get an AR for that. Get a good bolt action if you want range. For your $1,000 you can get a very nice Savage or Remington that will be more accurate in a round that is suitable for hunting. The 6.8 (I assume you mean SPC) is simply too expensive and hard to find locally, especially compared to a .270, .243, .30-06, .308, or pretty much any other mainstream, tried-and-true round. Also remember that less common chamberings tend to cost more, so your 6.8 SPC AR is going to be even harder to find for under ten Benjamins.
On the other hand, a decent bolt-action might only cost $400-500, so you'd have plenty of money left over to buy another rifle, or a pistol, or a shotgun... If the only reason you want an AR is because of the look, you should reconsider. Heck, even a Mini-14 Target BOUGHT AT THE MANUFACTURER'S SUGGESTED RETAIL PRICE would cost barely more than you'd pay for your AR, would fire the same .223, and would be quite a bit more accurate. It would also be far more reliable than a low-end AR.