IMO this is what you do . 300 yards is nothing to shoot . Almost any bolt gun will do this well . I shoot .80 MOA with my $360 Ruger American rifle @300yds .
That's why I recommend the Stevens action if you want to go the route of a build. I got my Stevens for less than $200 used, and even a new one is right at $300 or less. Add to that the Boyd's stock he likes and you've got a rifle that will shoot MOA accuracy with stock that he likes, all for $400 or less. And again, the nice thing about doing this with something like a Savage is that down the road you can convert the same action and stock to a caliber suited for longer distance for a few hundred dollars or less.
I'm probably $500 into this rifle including optics, mounts, and everything else you see and I couldn't be happier with how she's performed. I was out last weekend ringing 10"x10" steel at 600 yards no problem in decent wind shooting a Hornady 68 BTHP handload. The Savage Stevens rifles flat out shoot.