As stated, .308s are used regularly in F-class and Palma competition. They're usually built-for-purpose outfits. 1K is doable with a standard rig too, but it's rare that a 6" group will appear until you have ALOT of training & practice with the wind and solid handloading techniques. Also, as stated, you can use the .308 brass you have on hand to go to a more suitable caliber: 6.5-08 (260 Rem) or 7mm08.
That being said, a bone stock Savage F/TR rig might suit you. They have a long barrel that's forgiving of velocity, and they appear to shoot well out of the box. Loads are typically easy to tune, and you can get exceptional practice shooting F-class and tactical matches. YMMV.
6" with a 308 well it can be done! I just have not done it or seen it done but that's not that it can't. The first rifle I built was a 308 to shoot long range, I wanted to build this rifle myself so I went on the cheap (I thought anyway) A Mauser M-48 action, Parker Hale 1-14 twist I think it's .900 straight. The rifle shoots much MUCH better than I would have thought I don't shoot this at game very far I would call it quits at 600 yards not because the rifle won't do it I am just not that good yet! I have shot a 2 1/2 - 2 3/4" group at 500 but to do it again well the weather was just right and in the 6 years since I have not seen the conditions like that again. with the barrel twist I have I can't shoot the 175's dang it they tend to make some interesting holes in the target :-)
I just finished a 243 with a Broughton 1-8 twist to shoot the long range mainly pasture maggots and an occasional deer and if the weather would calm down a bit I might actually get out and see how it's going to shoot. I have heard very good things about the 260 for LR and some good friends shoot the 7mm-08 and they love it! they are giving me a hard time since I'm going 280AI for the next rifle. You say you have a mess of 308 brass? well sir you can form the cases for either the 260 or 7mm-08 from the 308 being that was the parent case for those wildcats at the time before they became factory rounds, forming those cases is not a difficult thing to do trust me I can do it so anyone can, I have formed some 243 from 308 just had to step it down and not try in one pass.
One thing I was thinking about you said yotes at 1K not saying it can not be done but I have yet to have them stay in one place long enough but either way I love my 308! but I'm thinking the 260 or 7mm-08 would much better suit you, these guys on here have a great deal of knowledge and a bunch of "Been there, Done that" I have figured out but you have to build what you want and the caliber choice I have found to be the hardest part of a build! you could always just get you a Savage action and a couple of barrels? I'm thinking that way on the next one. Best of luck to ya! Later,
I have a Savage in .308 varmint barrel. I built it and hope to complete it this spring and try to get it shooting out to 1000 yards for groundhog season. Thanks for the post....cause I had the same question.