I was in a similar situation with bird photography. One thing to keep in mind is the added AF sensitivity in the D600 (and the D800) that allows routine use of the 1.7x teleconverter on an f4 telephoto. I found the 1.4x teleconverter to be a very good performer on my 600mm f4 lens used with the D7000, like using a 1260mm f5.6 on a full-frame. The 1.7x AF performance was marginal, used only in a pinch. However, the D600 has improved this AF , to the point that the 1.7x teleconverter works very much as well as the 1.4x does on the D7000, turning the 600mm into a 1020mm f6.7 on the D600. This makes up for some (not all) of the "reach" issue.
I bought the D600 to get full-frame on scenics and "animal in the environment" shots. It is my first FX camera, and I suspect that it will deliver superior performance for that kind of application, if the DxOMark ratings are any indication. I am keeping the D7000 as a second camera, as the handling, U1 and U2 settings, controls and menu layouts are nearly the same, and the power accessories and IR remote and wired remote are the same. That was my primary reason for not going with the D800. The second camera will have my 200-400mm f4 on it, so there is the need to easily switch from one camera to the other without fumbling for a button that isn't there.
If you are getting rid of the D7000, and are willing to buy more accessories and train yourself for a new camera, the D800 is a better choice, as you get a DX crop that has nearly the same pixels as the D7000, (15.4 megapixels for the D800 in DX mode versus 16.2 megapixels on the D7000) PLUS you get the better AF sensitivity to make it even better for birds.