The most important measure of accuracy to hunters is the measure of how consistently close the first couple of shots - out of a cold barrel under varying atmospheric conditions - are to the intended target throughout the intended range envelope; repeated groups out of a warm barrel maybe good for the hunter's ego but really have no practical relevance. This is diametrically opposed to what is needed from the target shooters. According to the barrel makers I've spoken with there is no theoretical difference in "hunter's accuracy" (as I've described it above) between heavy weight barrels and pencil-thin sporter barrels. As for a varmit hunter-shooter I would think that the factors required of the target shooters, ie. the number of repeated shots over an hour or more, definitely is the most important factor.
Being strictly a hunter-shooter, I would think the way to approach the weight factor - short of a carbon-wrapped barrel - would be cutting back on length before going below a heavy-weight sporter contour. Currently I'm building a longer-range hunter in 280AI based on a fluted Bartlein 3-B planned for a minimum finished length of 26-inches. Compared to all my other hunter contours, it would be considered a beast so to speak. Although weight is not my primary concern, since I build my own composite stocks, I can continue to maximize weight economy through trial and error. In your case - assuming that you will be using a bi-pod - I would NOT shy away from light-weight hunting stocks as long as the forend is strong AND rigid enough to resist flex and withstand the continued radial stresses imparted by the bipod during hundreds of firings. To meet both those requirements, that ususally requires a continuous aluminum spine/chassis from the bipod mount back to (and including) the front guard screw - bottomline; that eliminates the "pounders", as well as the TI type stocks having only independent pillars, but just about all the manufacturers make full-chassis sporter-style stocks in the 2 to 2.5-lb range suitable for bipod use. Don't know whether my rambling opinions help or not - as always, YMMV.