You can use the Kenko TC by turning off AFMA in the camera body, or use the canon TC by inserting a extension ring.
Just for the heck of it, and since it is a sunny afternoon, I tried some different combinations with my 5D MK III and my 100mm L.
First, a short extension ring. That worked and so did autofocus, its a reasonable handheld near 1:1 solution.
Then, I put the camera on my light table mounted to a sturdy bolted down head.
I used liveview to try the tube, tube plus canon 1.4X, tube plus Canon 2X, and tube plus both TC's. In all cases, I was able to use live autofocus, but the image without TC's and just the tube was the clearest.
I also had a old non reporting 2X TC (Quantaray) which came with a used film body. It did not need a tube, live AF worked fine.
Then, I took the tube and Canon TC's outside into the bright sunlit outdoors to try phase detect AF. It was unacceptable in all cases, and could only rarely fix on the target. I tried both AI Servo and one shot.
Just the short tube worked
So, with a tube and TC, the camera will not autofocus in any soet of reliable manner. This is my second 5D MK III, as I recall, the first one worked the same way.
If you are doing live view AF, use a self timer, and it will be acceptable with a 1.4X, but just tubes might be better if you can get close enough. Manual focus works as well, of course.
Thank you for this detailed input. I think I'll forgo adding the expense of the tc. Remind me that if I ever meet you and your wife, I owe you dinner