Robster, ditto on the Mod 7. A fine short action.
They are not wildcats, but a .260 Rem or a 7mm Rem SAUM would be good candidates. The 7 SAUM would require a bolt with a .535" face and might be difficult to modify to feed. A 6.5 x .284 is a fine long range cartridge, but in the M7 would have to be single loaded, and an unfired round would almost certainly have to be ejected by pulling the bolt and round out the back of the receiver. On the plus side, the .260 Rem and the 6.5 x .284 would work with your .308 bolt, and the .260 would also feed without any modifications. Feeding wouldn't be an issue with the 6.5 x .284 because it could only be single loaded anyway.
I don't know the F-class rules, but if a round doesn't have to feed out of the magazine, the 6.5 x .284 would be my choice for a long range rifle in a short action.
Just my opinion, Tom