I have the 70-200/2.8 "Mark I" and both the extenders you mention.
I will occasionally use the zoom with the 1.4X II. Image quality is acceptible to me. However I have 300/4 and 300/2.8, so normally just use those instead and they both deliver better IQ than the zoom + TC. Both the 300s also work very well with 1.4X, and the f2.8 works very well with the 2X II.
I would expect your 100-400 to give better results at most focal lengths, than the 70-200 + TC would. By any chance, you don't have a "protective filter" on it, do you? The 100-400 is known to not "play well" with filters. So long as you have good light, I'd probably just try using that instead (I'm not a big fan of push/pull zooms, so don't have one).
I will not use the 2X II on my 70-200/2.8.... there is just too much loss of IQ for my tastes. I understand the 70-200 Mark II and 2X III are a better combo, but it's a pricey combo and I still see some people not infatuated with their results.
Try out the 1.4X II on your zoom. I think you'll find it acceptible. But that combo only gets you to 280mm, so you will probably still want to have the 100-400 handy.