I am an avid .308 Win. fan for deer and bigger game within 300 yards.
At longer ranges like what you are talking about velocity drops off rapidly and so does terminal bullet performance.
The super bullets will not cut it at those velocities.
Standards like the Hornady spire points will perform a tad better. Or.....the Nosler partitions.
I go to a more powerfull caliber for the above reasons.
The .308 is good up to 600 m. on people. Wounding is as good or better then killing in a military sense.
N_O_T in hunting, wounding should not be an option, neither should a cavalier attitude by taking big chances with a big game animals life and risky recovery as some others pointed out. Or ................to prove to yourself that "it can be done".
I am not accusing you at this point of calousness since you genuinly might not be aware of the limitations of the .308 in hunting.