I think you're going to have a hard time finding a custom action sub $1k. If you're firm on that budget, I'd just get a 700 action and have your smith blueprint it. Honestly, it'll shoot better than 90% of us can shoot if it's done right.
If you can increase you budget a few hundred dollars, THEN your options open up. I'm particular to Defiance because they will spec it out just about any way you or your gunsmith wants for your preferences/needs. If you're trying to squeeze as much accuracy as possible, I'd look for something that has an integrated rail just because it adds rigidity to the action.
Really, ANY custom action is capable of incredible accuracy. The differences are more about preference than accuracy. What matters, when it comes to accuracy, is the smith you choose and the work he does. If you use premium components (no matter the brand) and a quality builder, you're going to end up with a accurate rifle.
Good luck on your build!
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