I tried the 300 + 2x III both on the 5d III and the 7d, and the 5d III had no problems with quality or focussing. However, in a few instances the 7d would not even try to focus, until I had recomposed and tried again. This was maybe 3 frames out of more than 900, so nothing to worry about in my trial cases, but if you really absolutely have only one try for a shot, it does pose a small risk.
I admit that I have never tried with a 500, as the 300 is my first longer lens, but I would be inclined to say that the quality is ok. The first two with the 5, the last one with the 7 (at ISO 1250, actually a bit higher than I try to stay with this body).
P.S.: I also used the 300 with the 1.4 III and the quality and AF-speed is outstanding - and I didn't have any problems with the 7d either, so if you can afford to crop, this might be the safer way.