I think with a $3k budget you can get a nice entry level 6.5 CM. In turn-key, I would look at three brands:
1) Tikka (mentioned), which never really gets the credit it deserves. They make an outstanding action, and their rifles run from entry level to high-end tactial and competition. Personally, I like the Tikka barreled action in a Cadex frame, but that is another topic...
2) Bergara. These guys are new entrants into the rifle business, but their barrels are solid in the mid-market. Budget aside, I would pick a different barrel, but their barrels are damn good, and the rifles seem to be doing well.
3) Cadex. There is a new rifle out this year... or, rather a re-introduction of an old rifle, which, in the past had some issues. The Cadex team, in their normal engineering fashion, went back to the drawing board and have created a great entry-level rifle with a polymer-aluminum stock and a middle class barrel, along the lines of a Bergara barrel or Criterion, etc. All of which, as I have said, are perfectly fine barrels. Just not with the reputation and production methods of some other premium brands.
Finally, you might look at a good gunsmith and get a Remington 700 action with good barrel, and pop it into a Cadex chassis. In our shop, we are looking to do that, to find a great rifle at a great price. We are not there yet, but hope to announce something in a few months.