First, have fun! F class is a great, simple, discipline, and very rewarding.
Second, your AR is not a 1/4 minute gun, and you are not shooting 1/4 MOA - at least not in the context of F Class. 1/4 MOA wins many benchrest matches. The only way that is possible is if you are cherry picking your best groups, disregarding flyers, etc. In F Class, you have to shoot 20 shots (depending on the course of fire) in a row under 1 MOA to clean the target. That bar is a lot higher than a three shot group. A lot of 1/4 MOA rifles turn out to be 1 1/4 MOA rifles when the shots pile up. The reason I mention this is to keep you from getting discouraged. It's just statistics, not you.
At 300 yards, load up some 80 grain bullets as fast as you can get them while keeping five shot groups (and flyers DO count!) as small as you can - my guess is if you do this honestly, you'll have about a 1 MOA rifle - maybe a little better. But that is still good enough to shoot some good scores at 300. For 600 yards, I stop load development when I find a consistent 1/2 MOA (that is 10 shots in 1/2" at 100 yards every time). I doubt you'll achieve that with an AR, but that shouldn't stop you from trying.
You don't need a spotting scope to get started, but it won't hurt. What you will really want is a bolt gun with the longest barrel you can find and a 36x or higher powered scope. (This is for 600-1000 yards, you can get away with less at 300, which is far more forgiving a range - I know guys who wont shoot at 300 because they think it's "boring". I'm not sure I disagree). One of the nice things about F TR is that the equipment is pretty well defined and the specifics don't matter as much. You don't need a fancy action or an expensive scope. A Remington 700 or Savage with Weaver benchrest scope are more than good enough to win matches if they're set up right (long barrel, good loads, proper bipod, a stock with a level toe, decent trigger, heavy bag).
