Savage route, and what TRG said.
Just finished a Savage 7-08 build...
And just got an aftermarket .260 barrel to swap out on my son's .308 Savage.
Barrel wrench and go-gauge, made my own barrel blocks.
7-08 and .260 have very similar ballistics.
We shoot steel at long range, so retained energy is not a concern for us.
However- since you are building/buying for a hunting application, you should go with the 7-08. The 7mm bullet will have more energy downrange. And, it's also a good choice if you don't handload as commercial ammo is more available, and less expensive, than .260 which runs $30- $50 box
Hit the pawn shops and look for a "donor" action if you can find one...barrel condition, stock condition doesn't matter. Savage 10/110 , or Stevens 200 in any caliber with the correct boltface (preferably- although you can change the boltface on a Savage relatively inexpensively).
Should be able to find something for $250-$300. Sell the barrel and stock.
Aftermarket barrel ($250), stock ($200-$300), maybe even a Timney or Rifle Basix trigger. Especially if you can get $100 or so from the stock and barrel, you'll have built a very accurate rifle within your budget.