The Savage 12 F-T/R is getting a great reputation, friend of mine who shoots more Long Range than I do says he is seeing a number of them at matches and they do well.
If you want something fancier when you have worn out the .308 barrel, you can make it shoot about anything that will go in the port. Savage bolt heads are separate parts that can be changed out if you want a belted magnum. A 6 BR or 6.5x284 would use the standard head, though; and call for only a new barrel. (And dies, and brass, and cleaning jag, etc., etc.)
However, if you are shooting real NRA F-class, changing the caliber to something other than .308 (or .223) will put you in F-Open versus a whole new range of competition.