The 5D III will take a Kenko and my 100-400, but it seizes up or doesn't work with the 300 series II. If you want a x2, the native 2XTC will be much better than two 1.4s.
For amusement, I tried the Canon 1.4 TC with my old Sigma 10-20 f/4, which is equivalent to the EF-S but does not stick out at the back. Without the extender, it produces a circle filling about 2/3rds of the full frame of the 5D III. With the extender, it cover full frame to be a 14-28 f/5.6, and correctly reporting the aperture etc.