Lightweight Build from Factory Rifle

[Outback]

I need an 18-20" barreled .308 (or other round that would be effective in a shorter barreled rifle AND suitable for elk/Nilgai sized animals). I am looking for a factory rifle that I can just replace the stock and bed it - perhaps swap the trigger if necessary. I'm hopeful for a 0.5-0.75 moa rifle with handloads. Is this possible and what rifle would I begin with? Savage? Ruger? Trying to keep it low priced to fly under the CFO's radar.