Jason,
If F Class with a .308 is what you are looking for you have 3 options.
One is a factory gun called F(M) class. Few mods you are allowed like recrowning and mods to the orginial factory stock and a bedding job. Any caliber, any bullet weight.
Option #2 is F(F) class that will allow custom rifles in .308 with 155 gr bullets or .223 using 80 gr bullets.
Option #3 is F(O) open class that you can use any caiber, any bullet so you could shoot 175-200 grain bullets out of your .308. F(O) is a tough class for a .308 if you are shooting against guys shooting better rounds.
For a beginner rifle under $1000, look at Savage, Tikka, Remington.
Danny Boy,
Try the 80 gr bullets, then that 0.35 group may the consistent. With a higher BC you will see the difference at 300m for sure.