The 1.4xIII will cut AF speed by 50%, the 2xIII will cut it by 75%. I've used 600mm f/4L IS II with the 2xIII for BIF and the AF is fast enough, but only just - and YMMV, as I'm using a 1-series body (the higher voltage battery drives the lens AF motor faster that a 7D can).
As for IQ, the 2xIII will do fine on the 300 II - in fact, the combo is similar to the 600/4 MkI (although the MkII is better). I suspect the 1.4xIII would be a better choice for the 135L, though - that will take a bigger IQ hit from the 2x than the supertele.
I did the test you describe with my 600 II and 1D X, and the 2xIII delivered better IQ than the 1.4xIII cropped to the same FoV.
I haven't tried that with my 135L, in fact, I've never even used a TC with the 135L (probably because I've got the 70-200/2.8L IS II).