For avian photography you are nearly always using a zoom tele at full stretch so may as well use a prime. Zoom is useful if you are using the lens for large game animals as well, where 400mm fixed focal length could end up being too long. In a game/nature reserve situation where you will encounter a range of different animals on a daily basis the zoom is probably more versatile.
The 200-400 will have about the same reach as a 300/2.8 + 1.4TC combination. Both are f4. A lot of local game and avian photogrpaphers use the 200-400 with a 1.4TC to get an effective 560mm f5.6 lens and this is what you would have to do in order to perceive any difference to your existing 300 or 300 + TC combo..
The questions : 1) Is the 200-400 sharper than 300/2.8 + 1.4 TC? Possibly but I think the difference would be slight. 2) Is the 200-400 + 1.4TC better/sharper than 300 + 2.0TC? Unfortunately I don't know but our local avian photogrpahy gurus (www.outdoorphoto.co.za) are saying the new TC 2.0E III is much better than the older TCs so this is the cheapest upgrade if you are using an older TC.
The D800 has 15 AF sensors that will work at f8 so the use of 200-400 + 2.0TC is possible. This is the only way that the 200-400 is going to give significantly more reach than your current setup but again, I cannot say if sharpness will be acceptable.