I couldn't imagine using the 600/4+1.4x for BIF.
Not only can I imagine it, I do it routinely.
EOS 1D X, EF 600mm f/4L IS II + EF 1.4x III Extender, 1/2500 s, f/8, ISO 640
With the 2x as well…
EOS 1D X, EF 600mm f/4L IS II + EF 2x III Extender, 1/2500 s, f/11, ISO 1000
Wow! You have some great technique! Of course, I bet that 12fps and faster AF drive of the 1DX is immensely helpful, too!
The shot taken with the 2x is good, but the atmospheric blurring is generally why I probably wouldn't do that much. But seeing what you did with the 1.4x...if I ever get, or rent, a 1D X, I'd be willing to give it a try. I'm not sure how well a 5D III would fare. I know for a fact that my 7D doesn't handle the 600+1.4x for BIF very well...it's just too slow.