If it were me, I'd start with a Hog Hunter .223 and an aftermarket stock. I have done a lot with Boyds. The Pro Hunter is only about $100 and you can add an adjustable check for another $60 or so... Bed and pillar it and see how it shoots. For another $10 you can replace the trigger spring for a lighter one and get down to 1.5 pounds. You just might be surprised how well it shoots.
Next, I would replace the trigger with a Rifle Basix Sav-2 aftermarket trigger.
Then, I'd have Jim from Apache gunworks http://apachegunworks.com/pricing.html build me a SS heavy barrel, either low end blank or, for another $100 or so, have him build off of an X-caliber Stainless steel blank.
Somewhere in there I think you'll find a consistent .25-.50 inch shooter for a lot less than what you are proposing for a budget.
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