In DX mode the D600 will give you a 10.5 mpx image. The D800 will give you about 15mpx, which is almost the same as you get with the D7000.
In DX mode the D800 will give you about 5 fps. The D600 is rated at "up to" 5.5 fps in both DX and FX mode. Depending on the nuances of Nikon's official spec you may get the same frame rate while birding, where you are presumably unlikely to be shooting in FX mode. Your concerns about frame rate are probably unwarranted in the real world of highly cropped birding images.
The D600 or D800 (in DX mode!) will improve low light performance by about 1/2 to 2/3 stop in the range where you probably care most, per Bill Claff's numbers (the D600 numbers being tentative, presumably awaiting more data). An update to the D7000 would probably perform the same.
You will get very large files from the D800 in FX mode but that comes with any FX camera that is going to compete with the D7000 in pixel density. So if you generally like FX (you have a D3s) then you just need to get your server farm more up to date