I use a 5d3 with a 100-400 all the time. It is one of my favorite setups and I have used it extensively for wildlife, birds (including BIF) and even landscapes.
Once they upgraded the firmware to allow AF at f8 I ran a FoCal test with the 1.4x to see what the iq results were. It took 3 test runs to get a result. The program kept telling me it could not produce an accurate result. The IQ numbers it came up with were quite poor. In fact cropping an inage without using a tc would produce a far better result then an image with using tge tc. My 1-400 is very sharp and a "good" copy. My 1.4 is a version ii and has tested excellently on my 300f2.8 and 500 f4 just fine.
Not one to take the word of only a calibration program I took it to my favorite local haunt to photograph osprey, herons, red wings etc to give it a good field test. My conclusion is that this is an unusable combo.
The AF is terrible. It hunts and has trouble locking onto static subjects that fill half the frame. And when it does lock on, it is not always locked on. The images that were relatively close with AF were extremely soft at all apertures. F9,11,14 would not produce usable results. The time of day was sunrise for about 3 hours after on a clear day, so I had plenty of light.
Disappointed I ended up testing the 400f5.6 next with the 1.4x. I was actually very impressed with AF but the iq was not that great. Not horrible, but not great.
IMO neither combo is usable. For more consistently good results I have opted to not use a tc with either lens, and then crop in post. Hope this info helps.