Thank you so much for all the comments. Since my last post I have had a chance to use the 5D Mark III + 100-400mm + Kenko 1.4x combination quite a bit. I was initially hoping to shoot raptors but haven't found the time so I have instead been practicing on shore birds. In general the combination works very well and I plan to continue using it for birding when I am focal length limited.
You can, for example, see a sequence of 6 consecutive frames (all sharp and in-focus) shot with the combination at 560mm f/8 here:
Here is a closer view of frame 4 in the sequence:
Pros of using extender:
1) Images are still sharp and I get greater detail than without the extender.
2) All 41 cross-type AF points work!
3) Tracking sensitivity is as good as bare lens.
4) The subject fills up more of the viewfinder and covers a larger number of cross-type AF points on the 5D3 for focusing and tracking. This is actually a huge benefit that I didn't appreciate until I started using the combination out in the field.
1) Initial focus acquisition seems a tad slower than the bare lens.
2) Noticeable purple fringing at the corners. While easy to correct in LR4, it is still annoying.
3) Slight decrease in contrast.
4) Biggest issue is the most obvious one - now the combination is an f/8.0 lens, which really doesn't let much light in and won't blur the background as much as with a faster lens. This is the main reason I hope to upgrade to a faster lens when funds permit, but for now I'm happy with the combination.