I'd personnally look into purchasing the additional rifle. That way two of you could hunt coyotes together. The 243 is adequate for deer as long as you recognize it's limitations. The AR platforms fill a good niche, but I'd think you would have more options with a separate rifle.
I've struggled getting good groups in the Remmy 1 in 9 1/8 twist barrel with the light (70gn) varmint bullets. My dual purpose load for coyotes and deer with a 243 is a Sierra HPBT 85gn Gameking. 87gn Vmax's do great as well
A 400.00 remmy plus a 400.00 dollar scope (recommend zeiss conquest 3-9) puts you at 800.00. If you don't like the plastic remmy stocks look at Stocky's Gun Stocks
. They have good selection of wood/laminate/composite stocks.
Hope this helps and good luck