Trying to hit a dog at 500 yards without a rest is a hard thing to! I would go with the 308! But even with that you need to go shoot the gun at 500 yards to see how much drop you are going to have! It's going to be alot! The 223 will go out that far but the wind will push it around. I just hope you can shoot like Jerry Miculek to make a shot like that
