In the price range you're looking at, if accuracy is a key point, get a slightly used, bare bones Springfield M1A (should run about $800-900; new one would probably run between $1150-1400, depending on how much your dealer wants to gouge you). They aren't match rifles, but they're pretty accurate... maybe 2, 3 MOA out of the box.
Do not suggest a Norinco M1A. The receivers are soft. Yes, you can get the receivers heat-treated and up to a comparable level to the Springfield, Inc. receivers, but by the time you've gone through the trouble and expense of that, you'll wish you'd have just bought an M1A.
You won't get a 2 MOA rifle out of a stock FAL. Yes, you can throw a grand at a FAL and accurize it, but they really max out at 1.5MOA. Don't get me wrong -- I love my FAL, it shoots very well, and I'd recommend it to anyone who wants a good, reliable range rifle or SHTF rifle, but it's not an accurate weapon.