The 100-400L seems like it best fits all your requirements. Since you have a 6D, you can push your ISO to compensate for the slower aperture if light becomes challenging. While I do not use mine for field sports, I do use it for wildlife in motion, and the AF has been fine on a 5D3 and 7D.
I wouldn't worry too much about the rumors of the 100-400L MkII. As pointed out above, if such a lens does get announced it will be over your stated budget. Besides, a new version of that lens has to be the longest running rumor in Canonland -- I bought mine four years ago, and it was an old rumor even then.