I own a 20 inch 308 win and 22 inch 6,5×55 barrels for the same Blaser R8 system. I have used both extensively and have given a lot of thought about which one I prefer over the other over the years. The 6,5×55 with 22 barrel is the most accurate thing I ever shot and it's recoil is pleasant. It drops less at a distance than a 308 win. I really like it, I can hunt everything from fox to reds with it and I can do it with extreme accuracy and with nice low recoil.
308 win with a 20 barrel has a bit more drop, a bit more recoil and is not as accurate as my 6,5×55. But the overall rifle package is shorter, 308 manages shorter barrels better, it also balances better as the barrel is shorter and also lighter - considering the rifle bore is larger with 308 than it is with 6,5 mm. Both barrels have the same contour, but since the 308 win barrel has larger bore and is also shorter it comes out considerably lighter and better balanced when outfitted with a mod. The 308 win also packs more punch and is even very capable on driven hunts. I mounted a light Svemko Nano silencer on it (185 g) which does not make the rifle front heavy at all and Swarovski Habicht 1,5-6×42 scope which at 440 g is a very light and elegant scope. The whole package weights a bit more than 4 kg.
So since we are talking about a light short rifle I would give the overall edge to the 308 win. While I really like the 6,5×55, the 308 still comes out as more universal, shorter, lighter and more capable package, ideal for short woodland weapon.