As mentioned above, .308 isnt ideal for long range hunting, let alone being shortend to 16". Yes you can run a lighter pill with a stiff load of 2208/2206h behind it but it doesnt really pan out down range, you get the extra muzzle velocity but because your running a light & short but fat pill = low BC - it falls over pretty rapidly as far as trajectory & energy + gets pushed around by the wind more.
Your best bet would be something like pommy has mentioned, 168gr ELDM would be a good pick. But even assuming best case scenario, you might see around 2600fps with a really stiff load of 2206h, with the 168gr ELDM thats down to 1760fps @ 500m & over 2ft of wind drift in a 10mph cross wind. Not ideal for trying to accurately and humanely kill game, but a good shooter thats had plenty of practise & confident with his gear & processes could do this no problem.. Average wind call and things can get messy real quick.
Imo, If you dont want to change rifle/calibre, i would limit things to 400m rather than 500.
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