Weatherby Accumark or Remington Sendero is where I would look.
I own several Weatherbys, though not an Accumark. I really want one of those next. I do not recall ever hearing a bad report on the accuracy of one of these, and it is the lighter of the two choices. I absolutely love the 54 degree bolt throw and the smoothness of the Mark V action. I see that there is one used one on Gunbroker now, and I can't recall seeing one previously (but I haven't looked for a 308, either). I toss around getting another 7mm Wby mag or going for a 300 or 340 Wby mag...
I own one Rem 700, a Sendero. As others here will note they are very accurate for a factory long range rifle. However, not available in .308 which apparently puts it out of contention for you. And, it's a few pounds heavier than an Accumark.
You might want to look at the "Varmint SF" Remington if you're stuck on the .308 as your choice. It's the only Rem available that meets your criteria - unless you decide to have a custom rifle
built for you.
That's what I see as your choices.