My experience is that the Canon 300mm f/2.8 II 4 stop IS is much, much better in mode 1 at freezing movement than is the Tammy. The shutter lag of the Tammy is, accordingly, less bothering, and I have found it negligible for big birds in flight. I use mode 3 for the Canon. At 600mm, I need 1/2000 for the Tammy without IS or VC for it to be absolutely sharp 100% of the time.
And what shutter speed do you prefer for the Canon with 2X TC in mode 3, for BIF? 1/1000? I've found that's not fast enough for small birds, but I guess that's not saying anything that's not obvious.
Small birds are usually too fast for me at 600mm. Hovering birds of prey or lumbering heavyweights are what I can manage. And even for them the faster the better.
5D IV, 5DS R, 7D II, EOS-M, Powershot G3 X, 400mm DO II, 300/2.8 II, 1.4xTC III, 2xTC III, EF 1.8 STM, EF 70-200/4 IS, EF 24-105, 15-85, 100-400 II, Sigma 10-20, Sigma 150-600mm C, EOS-M5 15-45, f/2 22, 11-22, Samyang 8mm f/2.8 fisheye