Seeing as you already have the 70-200 f/2.8L II covered, it would be cost effective to go with the 2x EF Extender III.
During one of my visits to a Formula One race, I shot back-to-back using the 100-400 and the 70-200/2x combo. I found the two configurations to be comparable in both IQ and AF performance. I will say however I much prefer carrying the 100-400, for it's lightness and compactness (1380g), over the prominently protruding 70-200 + 2x (1815g combined!).