I'm a remington guy as well, but to say what is the most accurate rifle under 1200.00 is like asking which came first the chicken or the egg? It all starts with the shooter, if you don't have good ranging ability, good shooting habits, then having a $10,000 rifle won't make a darn.
I know of 2 1000 yrd guys that are awsome, I know my range is best kept under 400 yrds if I want a resonable amount of dead coyotes, so I use what gun I feel the most comfortable shooting and one I shoot with all the time, I know I'll never be a 1000yrd sniper nor do I have the desire to be so, I call most coyotes into my range and the remington varmint 22-250 with 53gr Hornady V max's turns there world upside down. I would say the 2 best things a guy could get is a Harris swivel bipod and a rnage finder to improve accuracy alot with less money/time spent on a new gun. The guns made today and the ammo and powders we have, most name brands will shoot plenty good enough. I get 3/4" groups with my Rem and that is pleny good enough for killing coyotes.