I have the 200-400, and had a D300 now have a D800. I agree that the choice in the 200-400 vs. something like a 600mm is very different.
To me you get the 200-400 for the zoom flexibility in that range. It's a stellar lens that is sharp throughout its range even wide open. I know some will find fault with it at distance but they are usually comparing it to a prime at the same distance, and then you have to look VERY closely. While not exactly light it can be hand held, and it focus down so close that a large butterfly can fill a DX frame so you have a light macro lens with huge working distance.
And the VR-I version is darn close to as good as the new one so you might find a decent deal.
It's a high quality super zoom, but it's a lot about the "zoom".
I don't have the 300 or 600 but I do have a 200mm/F2, and I suspect the 300 behaves similarly -- TC's work wonderfully on them. On the 200-400 not so much. They are OK, in fact the 1.4x is terrific, but the 2.0 (at least the prior version I tried) was soft enough that I would rather crop and get big pixels than use it, so I sold it. So don't expect the 400mm end to give you 800mm without some compromises. And you may find it hard to focus the D300 on the 200-400 with the 2x due to light (mine would focus but not reliably).
I have done my share of birds, and I think if I had a 300/F2, I'd skip the 200-400 and consider a 400/F2.8 or 600mm/F4. Frankly I'd be tempted by the 400/2.8, it has a reputation as one of the sharpest lenses available and also for holding up very well for TC's. You also get the extra stop at 400 which never hurts. But I don't have a 300, so given you already have the 300 I'd probably try for the 600. But tough call (that and I want a 400/2.8 for other reasons, so I'm biased).
Probably should have asked "what sort of birds", one where you have time to frame and change TC's or not. But I tend to think of most birding as one where a zoom is nice but not a necessity. But reach and sharpness are, and wide aperture is both great for low light but also separation.