Are you intending to hunt w/the rifle and carry on long stalks or just use as a prone shooter...W/ 10-12 Hundred, i would probably go w/your initial suggestion and shoot it until you get a sense of what you like and don't like about it. As for alternatives to the SPS, you can look into a Tikka T3, Savage FP10 or Howa (not sure how the Howa is priced relative to the other two but shouldn't too far out of your budget, if at all).
Also, consider what bullet (or bullets) you want to run. This could mean anything from rounds in the 140-150 grain range to 190's and heavier...Twist rate will be crucial for heavier loads so ensure you plan for that (10 twist is great for .308)...The first thing I would do, modification-wise, is the trigger. I use a Rifle Basix in my .308 bolt gun and it works great...Depending on the rifle you buy, you can also have a gun smith adjust the trigger for you as well (usually around $50-$75 for a trigger job). I think i paid around $125 for the Rifle Basix about 4 years ago.