In terms of sharpness, the 100-400 and the 70-200/2.8 II are about the same. They are also similar in terms of AF.
If you're getting paid for this and are on a budget, I would recommend the 400/5.6. I honestly have no experience with it - though I do own its sister lens the 300/4. I would expect it to be much quicker than the first two in terms of AF.
For sports shots AF speed will be your primary concern. If you can, the best option would obviously be to rent a 400/2.8.
Note that I have noticed that on the 5D3 my 70-200/2.8 II + 2x III does seem to AF faster. I would still not call it a speed demon, but I did manage to get this shot with it - http://500px.com/photo/6268179
Again people, myself included, express their wish for a decent 400mm solution offered by Canon. I wished desperately Canon would listen carefully. So, I keep hoping for an upgrade of the 100-400 in the medium price range.
To be fair, Canon has the most awesomest 400mm solution of them all, in the form of the newest 400 f/2.8. And the other 400 f/2.8s are all pretty darned nifty, as well.
Of course, if you want long, fast, light, and cheap, you'll never be satisfied. But they've got the long, fast, and not-quite-so-absurdly-heavy market all to themselves. Everything else anybody offers will be a compromise of some sort...it might be lighter, but it won't be as fast and the IQ won't be as good, though it'll probably be cheaper (unless they go the DO route). Or, it'll be as fast and lighter and cheaper, but not as long...and they'll call it the 300 f/2.8. You get the idea.