I'm pondering the same decision right now.
I don't own either lens yet, but I will be getting the 70-200 2.8 IS mk II soon. I have the non-L 70-300 IS right now, and I routinely shoot at 300mm. I would not object to having even more range. 200mm won't cut it for what I do - so I'll either go with the extender 2x mk III or the 100-400. Obviously, the former solution is cheaper, but if I have to save up a bit longer to get the 100-400 in addition to the 70-200 2.8 IS mk II, then I will.
I understand that the speed of the focus drive is cut by 75% when the extender 2x is used (and by 50% when the 1.4x is used). I don't do a lot of sports or fast-action shooting (at least not in the 300mm + range), so I can probably live with this. It sounds like with the (mk II) lens + (mk III) extender at 400mm, IQ is on par with what you get from the 100-400 lens at 400mm. Yes, you lose 2 stops of light with the extender 2x, which means that either way you go, you will have a maximum aperture of f/5.6 at 400mm. So, this is really looking like a true "toss up" here.
I firmly believe in buying lenses with the native focal lengths that you really need, and as I've said, I'm not opposed to saving up a little longer to get the 100-400mm. BUT, here is what has me leaning in the direction of the 70-200 2.8 IS mk II + the extender 2x mk III, instead of the 100-400:
(1) The "drawbacks" in terms of sharpness, IQ, contrast, etc... inherent with any extender are offset by the fact that you're starting with a MUCH higher quality lens (the 70-200 2.8 IS mk II);
(2) Any issues with contrast can certainly be corrected in PP and the images can also be sharpened a bit in PP, if necessary. I’m willing to do this work, if lens + extender ends up being the better way to go;
(3) The 100-400 isn’t (fully) weather sealed, while the 70-200 2.8 IS mk II is weather sealed (as is the extender 2x mk III). I don't deliberately subject my gear to adverse conditions, but it is nice to know that my investment is protected, if the elements catch me off guard; and
(4) The 100-400 only has 2 stops of correction via its IS system, vs. 4 stops of correction via the newer implementation of IS on the 70-200 2.8 IS mk II.
Point # 4 above is what REALLY has me leaning in this direction. I don't lug a tripod around with me everywhere. 99% of my shots are hand-held. DOUBLE the correction (from a more updated IS system) has me thinking that the 70-200 2.8 IS mk II + extender 2x mk III might actually be "better" than buying the 100-400.
Am I crazy for thinking this?!? As you can see, I'm aware of the slower focus speed, IQ issues, etc. of going the lens + extender route here. I didn't see the issue of a much better IS system being discussed in this thread and I just wanted to bring it up. I'd love to hear your thoughts about this and anyone's actual experience as it relates to the two IS systems here. Thanks!