When you say "most accurate" you are setting yourself up for failure.
Because while there are plenty of factory rifles, and even custom rigs, in your price range that are guaranteed to do sub MOA groups, the "most accurate" one will vary from day to day and ammo lot to ammo lot.
The easiest way to get into the accurate gas gun is to get an LR-308. They usually do sub MOA with match ammo.
If you want a custom solution you can contact GAP or Patriot Ordnance and have one built for you. But I recommend the LR-308 because unless you are a better shot than someone who normally asks about "the most accurate semi-auto 308 under 4K" you should try to spend around 1.5k on the rifle, 500 to 1k on the optics, and the rest on ammo.
Jimro