I have to agree that an actual rifle "build" (starting with an action, barrelling it, stocking it, trigger. etc.) is probably not going to be a reality. If you want to tweak on something for the fun of it I'd start with a Mauser large ring action and go from there. If you want to make moderate modifications to an existing rifle to make it YOURS, I give up my opinion to the Remington 700 LTR. I know guys on here have affiliations with the Army sniper program or military r&d and contracting and talk about Remington not being up to snuff for them, but my LTR is absolutely the most accurate gun of any caliber I have ever fired. I have the .308 version with a 20" fluted barrel. At 20" you'll probably not make the 1000 yard club but the shorter tube and fluting helps with barrel harmonics. The only reason 1000 would be harder is the small loss in velocity will kick in sooner as the heavier (175gr+) bullet won't stay supersonic way out there. Some will say other rifles are better for the money and I can't prove them wrong, but the LTR is my choice. I over paid for the rifle itself only because I didn't want to wait- $950. I put a Leupold VX3 4.5-14X at just over $650 on it in Redfield rings and bases. I lapped the rings as well. Given the extra money I wold have gone with a Badger Ordnance ring and base set up as Dillinger mentioned. Easy 3/4 MOA groups using steel bullet Hirtenburger ammo. I haven't even started handloading for it yet to dial in a load and only have 86 rounds down range. Good luck.
"People live too long, dogs don't live long enough" - FTF Member-