I haven't used the 300 II (yet - it's on my short list of future purchases). The increased AF precision of the MkIII extenders apply only when used with the MkII supertele lenses (and possibly the 70-200 II, not sure about that one).You definitely need to keep the 300 on your list - it's a much more manageable size / weight than it's big brothers. I found this "poor man's 600mm" with the 2x very hand-holdable as well.
CR member AlanF routinely uses the 300 II + 2xIII with a 5DIII, you might check out some of his posts.
I keep wondering about the 70-200 II and the MkIII extenders, too. There are conflicting statements about this one from CPN, Chuck Westfall, and other Canon sources.