200-400 f4 and a 1.4 tc would be one combo worth considering.
80-400 vr is another lens that does very well but won't run with a TC. It does make a killer dragonfly lens on extension tubes though.
I have the 600 f4 manual focus. It's a great lens but you'll never be satisfied unless you only shoot birds that aren't moving. It also weighs a ton!
I got a second generation 500mm f4 AFS lens that's my favorite pure bird lens and was right in your budget range. Since I got the 500mm (and a 1.4 tc) I have not touched the 600mm MF lens for anything but moon shots. I can carry and shoot the 500mm off hand for several hrs. I can hardly lift the old 600mm, can't carry it all day either..
For long walks/day hikes, the 80-400 VR is hands down my favorite. It's a near perfect "park" lens. The 500mm is better in more controlled situations like a blind. You can walk and shoot the 80-400 all day, the 500 really starts to wear you down after a few hrs and forget about walking with the 600mm.
For my money:
Forget the 600mm MF for a pure bird lens.
Look hard for an older 500mm af-s lens.
Don't under estimate the old 80-400mm VR.. Slow focus but will still hang with a bird in flight pretty well.
I'd love to own a 200-400mm F4 but have no experience with it.
Lot of examples from the 500mm out
HERE. There are a few shots at the bottom of the gallery (jays and buzzards) shot with the 80-400mm.
Oops.. see already got a lens. Should have read everything. Maybe useful info for someone else..