The 400/5.6L was my main birding lens for over a decade. The 800/11 has been my main EIF (eagles in flight) lens beside the 100-500 for three seasons. While the 400 was very good it was not at the 800/11 level, perhaps not that far behind. I had the EF 2X II, but never used it because it was simply bad. Perhaps the v.III would perform better, but it will never bring the 400/5.6L up to the 800/11 level. It might be good enough though.
I have only experience with the one 400 I had which did well on the 18Mp crop sensors of the day. The 800/11 does very well on the much more demanding 32.5Mp sensor of the R7.