if you want to compete, decide what style of comp you want, then build a rifle to suit. If you like belly shooting LR, F class is probably your best bang for the buck. The TR section is growing leaps and bounds so you will have many to shoot with. The 308 is a required cartridge.
If you want to shoot using a sling, then Palma might be your thing.
For bench rest, to be competitive these days, you are going to need good tech and that means custom everything. www.6mmBR.com
for all sorts of winning combos for all sorts of different disciplines.
In light rifle classes, the 6mm and 6.5 pretty much rule the roost. In heavier classes, the 30 and 338 dominate but the 7mm in coming on very strong.
You have a fair budget to build a rifle but you also need to budget optics and reloading gear. Then there is this thing called practise, practise, practise.
If thinking club matches that are based around factory rifles, the Savage is an ideal platform. A heavy barreled rifle with the accutrigger will give you lots of trigger time which is really what will help you improve your shooting.
I have shot several factory rifles recently and they have all done 1/2 MOA or better way out there. Was at a 1000m fun shoot recently and a box stock Savage BVSS in 243 with handloads was making 11" groups in horrid winds with a beginner shooter.
At distance, doping conditions is the main weakness. As long as you have a rig that you like and can afford to shoot lots, that will teach you more then a full meal deal BR rifle you shoot twice a year.
Personally, I would get a Savage, bed it, tune or replace the trigger, spend the extra money on good optics.
If I knew what I wanted to compete in but don't need national winning gear, I would get a Stevens200, swap in a Rifle Basix II trigger, a suitable stock and a Pac Nor/Shilen prethreaded and chambered barrel in something competitive and fly at it. You will have invested less then $1000 for a very competitive rig (think 1/4MOA at distance).