I agree with the guys above. As much as I like Handis, they aren't designed for that type shooting. Sure, you might get lucky, but building a rifle for 500-1000 yards is different from building a hunting rifle.
With the .308 you're going to have problems at the sonic barrier, like Nick talked about. When I was in the Army (back in the dinosaur days), we considered the .308 an 800 yard cartridge. When a bullet crosses the barrier from super-sonic to sub-sonic, it does weird things, meaning that the things it does are not predictable. For maximum accuracy you want to keep that bullet super-sonic all the way to the target. Sure, some of the Army and Marine marksmen in the sand have been doing wonderful things with the cartridge and platform, but I think that if you talk with them, they don't consider the cartridge to be a reliable 1000 yard cartridge.
Don't get me wrong. I like the .308 cartridge, but just as Clint Eastwood famously said, "A man's got to know his limitations."
I've got a .308 Ultra that shoots very well, and I haven't done any load development for it. However, I don't consider it a 500 yard rifle.