135mm all the way. I use a 5dmk3 with a 70-200mm2.8 in low light levels all the time. With action you want all the speed you can get. Btw- the 135mmf2 has nothing to fear from any 70-200mm for portrait work. Honestly, sometimes the advise given out is very odd.
For volleyball, the 70-200L 2.8II would be my choice. Focus is nearly instant on a 1Dx and the zoom would be a must for me.
I shoot a lot of actor/performer headshots, and while I wouldn't give up my 85LII or 135L for portraits, the 70-200LII is also a great portrait lens...distance to subject to background being key factors.
I'm not saying the 70-200 can't do portraits, it can just fine. but the guy above said to get it over the 135mm in part because then you'd have a "killer" portrait lens. wells that's just silly. the 135mm would have to be considered a better portrait lens.
I'd still take f2 over f2.8 in a dark venue, zoom or not(actually i choose 1.8 over either). i bet the 70-200 does focus fast on a 1dx, but the OP is looking at a 6d, so i doubt that matters much to him. i haven't used a 1dx or a 70-200IS V2, but i have a ton of exp with the 70-200mm 2.8 with many camera bodies (not the 1dx). it's AF is pretty good but the 135mm is better.