A 2x teleconverter would work on the 70-200 f2.8 making it 140-400 f5.6. I'm not sure if the 70-200+2TC+60D would better the output of the 5D3 and sigma 150-600. Although I don't know for sure as I'm not a canon user I've never been 100% happy with the Nikon 70-200 with 2x TC (i was only ever impressed with the 2x TC on the 300mm f2.8). It partly depends on the conditions you are out in, if it's gloomy the 2x TC would likely struggle. My gut feeling is the 5D3 and 150-600 would be my go to combo for birds (by birds I'm guessing you mean owls or small birds rather than say puffins on farmed or Skomer. If it was Skomer then your 70-200 on 5D3 would be perfect, they get so close they can be within minimal focusing of the 70-200 which is around 1.4m on the Nikon).
Teleconverter are tricky above the 1.4TC, one persons 'it's great/ok' can be very different to another persons. It's best to try it for yourself. One option would be to hire one, another is to purchase used at the right price allowing you to sell on with little loss if it doesn't work out.