AF is very slow, but it happens.
That's also weird. Again, with a teleconverter that's behaving properly, the 1.4x cuts AF speed in half, the 2x cuts it to 1/4 of the normal AF speed. I'm not sure if that is a decision taken by the lens, tc, or camera though (the decision to *try* to AF is taken by the camera from the information given by the lens/tc).
So maybe because your lens is old, the III-series TC is getting confused, it's still AFing slowly because it's a 2x TC, but the camera thinks the lens/tc combo is still f/5.6 for some reason. What does the EXIF say? Take a few wide-open, then stop down a step or two and make sure it all follows.
Or, your 5D3 has a firmware bug. If that's the case, don't fix it...
We'll probably have to wait to see what results other people get (my dad's cousin has a 100-400, my mate has a 1.4xIII and he's getting a 5D3 in HK in a month or two, that's the closest I can get to testing under the same conditions...