You will get a lot of different opinions on this one.
For me, I like heavy for caliber bullets for bear hunting. In a factory load my choice would be a 180 grain Winchester Power Point. In a handload it would be a 180 Hornady Interlock. I shot a Maine bear with a 180 Hornady out of my .300 Savage, which would be around 200 fps slower than that .308. That bear went about 10 yards after the hit and was done. It was loaded with a Federal 210M primer and IMR-4064 powder. That combo would likely work in your .308 as well.
Lots of people will tell you that you need some type of premium bullet for bears and if that's your thing then fine, but with a 180 grain bullet at .308 velocities I don't think so.
Congratulations on drawing that tag and best of luck to you.