I tested this for you. I have shot:
70-200 2.8L IS USM II @200 f5.6 1/250th
70-200 2.8L IS USM II + 2.0 ext @400 f5.6 1/250th
70-200 2.8L IS USM I @200 f5.6 1/250th
70-200 2.8L IS USM I + 2.0 ext @200 f5.6 1/250th
100-400 4.5-5.6LIS USM I @400 f5.6 1/250th
100-400 4.5-5.6LIS USM I @800 f11 (just for fun) 1/250th
IS truned off and mounted on tripod. Off-camera flash used as available light is low.
I added a temporary gallery for you to look at with exif data intact. I actually think the 70-200 II plus extender outperforms my 100-400 @ 400.
I for one would be happy to leave the 100-400 at home and go out with the 70-200 mkII plus 2x extender.
Sorry it is not real-world test, I am moving apartment so lucky I managed to do these (plus I am busy with work!).
Note: I was worried about the result with the MKI 70-200 plus extender so I tested again, with the same result. It did improve with the liveview AF (LVAF) but I thought it unfair to use as LVAF was not used for the other lenses plus it only improved slightly.
edit: you can download the jpeg files: http://www.dpinparis.com/camlabs/camlab70200test/jpgs/
edit2: removed incorrect statement regarding use of 2x extender as pointed out by popo.