Okay, I quickly read the replies but mostly to see if the OP had given any more info on what he shoots and it looks like he hasn't yet. Therefore, all I can do is sum up the pros and cons as I own the 70-200 f/2.8II, 300II and 600II and TCs. So I think I can provide some good background info and if the OP has the time to reply to his own thread at some point then I may be able to help a little more with specifics......
First off the 70-200 with a 2.0TC (and I have the latest versions of both) is pitiful for AF speed. My 100-400 runs circles around it at 400 f/5.6. The 300II with 2.0TC is phenomenal at AF and the hit is there but it doesn't affect shooting. I just finished using it over the holidays to shoot small chickadees and songbirds handheld through lots of branches and the AF on the 1DX and 5D3 is spectacular (other bodies may not perform as well). The 600II bare lens at 600 f/4 is faster at AF and the IQ is a bit better than the 600 f/5.6 II IS(as I like to call it) but the IQ of the 600 f/5.6 is still pretty remarkable. No lens made by Canon, Nikon, or others in the 300 and above FL can touch the bare 300II for IQ and AF. That combo is simply unbelievable. The 420 f/4 IS II (300II + 1.4III) is also pretty fricken amazing.
The 840 f/5.6 IS II is also very, very good for both AF and IQ. The 1200 f/8 IS II is also very, very good for IQ....BUT, AF does take a hit because of the f/8. I have great shots of large birds in flight (eagles and swans) but anything smaller is a lost cause. For perched birds, the combo is phenomenal however.
If you really need 600 and want even more reach at times and can handle the bulk of the 600 then just go and get it. The weight is a bit of a problem (I handhold about 75% of the time and use a gimbal for certain times) but the real issue is the actual size and not the weight. This is where the 600 f/5.6 wins hands down as it is much smaller and lighter and I can handhold it for hours without fatigue.
These are my initial scramble of thoughts.....IMHO....YMMV
Posted below...Junco with the 600 f/5.6 IS II and Bald Eagle with the 1200 f/8 IS II both handheld.