I like my 100-400, but it's not the best lens out there. The IS is only 2 stops. The AF speed pales compared to the 70-200 f/2.8 II. The push-pull is a little awkward at times, and when extended, it can be awkward to shoot when hand-holding. Sometimes it has weird bokeh, too. That said, it is very versatile.
I haven't shot the 400 f/5.6, so I cannot comment on this lens.
The 70-200 f/2.8L II + 2x extender may be an interesting option. The AF on the 70-200 with 1.4x extender (280mm at f/4) is faster than the 100-400 at 280mm, and the IS is much better. If you go this route, you can always shoot it without the extender, which adds versatility. I haven't shot it with the 2x extender.
An older 300 f/2.8 with an extender may be a good option. An older 300 f/2.8 without IS costs significantly less than $3,000 USD, but a clean IS model (version 1) will cost $3,500 to $4,000 USD.