You will gain little going with the 12 benchrest, the 12 benchrest stock is fairly similar to the F class stock. The torque problem is mostly caused by the height of the barreled action and how it sits in the stock. The lower the action sits in the stock the less torque will affect it, the higher it sits the more it will affect it. Both the 12 benchrest and the F class the stock sits very high in the stock. Unless you are limited to a factory stock, I would suggest you keep the gun you have and change stocks. Something else to consider is Jack Neary (One of the NBRSA/IBS top shooters.) stated in the tuning video, if a gun doesn't track well, it can mean that the round is out of tune. I found this to be true, and very noticeable when I was doing load development on my 30BR. Something else to consider the more recoil a round produces the more it affects the way the gun rides the bags.
I would also suggest unless you are limited to the caliber to look for something else, there are far superior calibers to the 308, why handicap yourself? If I was wanting to shoot a pure factory class in 600 yard benchrest, there is other guns I would look at. First I would look at the 12 LRPV in either a 8 twist 6mmBR or a 9 twist 22/250. I would also look at the VLP in 243, or a 9 twist 22/250. Last I would look at the 12 long range precision in either 243 or 260. Get your self a good rest, with a number 2 bag, and these guns ride well. Out of all the rounds listed I would pick the 6mmBR hands down over everything else. Look at the Records just about everything is won and records are set with the 6mmBR or a variant of it.