If the OP says the 308 kicks too hard, it kicks too hard for him. Don't tell him how it doesn't kick, he says it does. This is the famous "perceived recoil" issue.
Several things I can see here that could help. If you want to shoot big game, you are going to have to accustom yourself to recoil. Hunting rifles are lighter than target or varmint rifles becasue people generally do not want to carry a 10 lbs rifle into the ountains or on a long hike. Lighter rifles kick more than heavier rifles, it's simple physics. This is a common problem with new shooters, they are unaccustomed to recoil, and want a do-all rifle. Fact is, there are few do-all rifles in real life: rifles that are pleasant to carry are not always pleasant to shoot. If you are stuck on shooting a 308, you could buy a heavier rifle or add a muzzle brake. Many people (myself included) do not like shooting around other shooters who have muzzle brakes on their rifles due to increased muzzle blast (redirected muzzle blast is actually a more accurate term), but muzzle brakes do reduce recoil for the shooter.
If you are not absolutely sold on the 308, a 223 or 22-250 would be a good alternative. For 200-500 yds shooting, a 223 will be accurate and serve just as well as the 308 with less recoil. I would opt for a heavy barreled model if you are just going to be shooting from a bench.