I have done just as well in terms of IQ with rabbits on the Sigma 120-300mm f/2.8 OS + Canon 2X III combination. Hard (if not impossible) to judge a web size image like this though - the fur on the rabbit's chest (the lighter patch below the chin) looks like it could be really great, and focal length could be something. The coarser fur of rabbits makes it a bit hard to judge sharpness - that could eb amazingly sharp but looks not exactly like it. Other than that I do think the bokeh is a bit nasty and the pic could have been taken at a slightly faster shutter speed to lower some of the eye-catching brown highlights in the grass.
I've been able to get sharp shots pretty reliably at 1/50, and even lower down. I know that I've gotten some sharp enough shots at 1/8 and even 1/5 with the bare lens.
It does seem that that 70-200mm II is sharper in the corners, and sharper (a bit) throughout (not really visible in shots without the TC however) and if you are more interested in wide as opposed to tele photos for your everyday shooting it makes more sense. Unless you have a real need for a longer zoom lens the 70-200mm is your choice and unless the 400mm focal length turns out to really limit you, staying with what you have seems the right choice by default. There are many times I wish the 120-300mm f/2.8 OS had the weight of the 70-200mm f/2.8 IS II - it's about twice as heavy.
70-300L I would forget about - part of the reason I avoided it was because it has nothing in the way of autofocus capability on non-1D series bodies because the effective aperture becomes too narrow.
My personal thought is that at $3700 the Sigma 120-300mm OS + 2X Extender III combination ain't cheap at all, but where else do you get a 600mm-equivalent telephoto zoom with f/5.6?