FYI, adding the 1.4X to the 300/4 causes virtually no slowing of AF. If you can tell any difference in the field, especially using the lenses on a 1D series camera, I'd be impressed. Personally I've used the combo mostly on 50D, 7D and more rarely on 5DII (it's a sports/action/wildlife lens combo, and the 5DII simply isn't a great choice for that type of shooting). I don't notice any perceptible change in AF performance on 7D.
I have not extensively compared the 400/5.6 (only used it a little, never owned one), so can't compare AF speed... But the 300/4 is nearly instantaneous under all but the worst lighting conditions.... So I can't imagine much better. The 300/4 isn't noticeably slower focusing than 300/2.8 IS and is probably a smidgen faster than 500/4 IS, both of which I do own and have used for years, so can compare.
The nice thing about the 300 + 1.4X is you have two focal lengths to work with, rather than just the longer one. Plus you get IS, so it's a bit more handholdable.

300/4 IS with EF 1.4X II teleconverter at f5.6, 7D at ISO 400, 1/320 shutter speed. Handheld. Available light. The down side is a little higher cost to get both the lens and the TC, and there's a slight bit of image softening, particularly wide open (f5.6) when the TC is on the lens.
Yes, the 300/4 is probably the closest focusing of all of Canon's tele primes longer than 200mm. This was cropped a bit...
300/4 IS at f4, 7D at ISO 800, 1/1600 shutter speed. Handheld. Available light. Pretty close to MFD. But, seeing that you're using a full frame camera, 400mm isn't really all that long a focal length for birding. Especially not with smaller birds.
500/4 IS with EF 1.4X II (700mm effective) at f5.6, on 30D at ISO 200, 1/1000 shutter speed. Tripod with ballhead and gimbal accessory. Available light. 500/4 IS with EF 1.4X II (700mm effective) at f5.6 on EOS 10D at ISO 200, 1/200 shutter speed, 550EX fill flash, tripod with ballhead & gimbal accessory.Whatever lens you get, you might consider adding an APS-C body such as the 7D sometime. That's like getting a 1.6X teleconverter with no light loss and a minimal AF performance penalty (how much depends upon which crop body is used).
Sometimes we get lucky, though, and 300mm is long enough focal length with birds...
300/4 IS at f5.6, 5D MkII at ISO 800, 1/8000 shutter speed. Handheld.300/4 IS at f5.6, 7D at ISO 1600, 1/500 shutter speed. Handheld, available light. Pretty close to MFD.300/2.8 IS at f2.8, 30D at ISO 200, 1/2000. Handheld, available light. About a 50% crop.300/4 IS at f5.6, 7D at ISO 3200, 1/640 shutter, handheld, availble light.