Thanks very much for all of the suggestions, people. Some really interesting ideas I'd not thought of - like buying a Canon 7D and using the crop as extra reach. I must say, a second body would be quiet useful at sports and other types of events when the moment can be lost while changing a lens.
I've been really impressed from the sharpness I've seen from the 400mm prime. I think my plan is now to get hands on my 5D3 and try a converter on my 70-200mm. If that isn't sharp enough then I'll try out the 400mm prime and see how it goes with that reach and no IS. If it's crying for IS then the 100-400mm will be the way to go.
These things are never that easy.
I'm also having to trade my 10-22mm EF-S for a 16-35mm. Perhaps the cost of this upgrade as well as getting above the 200mm range is actually making the 7D seem like an even more attractive option. I could use my 5D3 with my 24-70mm then either have my 10-22mm or my 70-200mm on a 7D.