Frankly, your best bet would be to buy a crop sensor camera and use your 100-400mm on it. That would give you better image quality and better performance than any available teleconverter combined with either of your lenses.
Using one of the cropper cameras is like having a "free" 1.6X TC on your lens... no loss of light slowing or defeating autofocus (your 100-400mm with any 2X added becomes and effective 800mm f11 lens that will not autofocus on your 6D). Your 100-400mm on a 70D or 60D or 7D would perform like a 160-640mm f4-5-5.6 lens would on your 6D.
I also agree it makes little sense to use a 2X on a 70-200/2.8 II, when you already have a 100-400mm lens. You'll end up at exactly the same place, but with lower image quality.
70-200/2.8 Mark II reportedly works pretty well with Canon 2X Mark III. Some people are happy with it, others are not. I wouldn't expect as good image quality with 2X Mark II TC on that lens... And I won't use a 2X at all on 70-200/2.8 Mark I, due to too much loss of IQ.
EDIT: great minds think alike... I just saw Lloydd's response.