Given that you have the 5D MkIII and 100-400, your cheapest and lightest route to a longer lens is an extender like the Canon 1.4x. That will still give you AF (with the center focus zone) at f8 and you'll be at 540mm at full zoom.
You can get a used MkI for under $100 or a used MkII for under $200
from Adorama. A new
MkIII extender will cost you $429 .
However looked without regard for you current gear, and assuming you wanted maximum reach, I'd have suggested anEOS 7D MkI or EOS 70D and a Sigma or Tamron 150-600/5-6.3. Probably the Sigma because it will take the Canon TCs (using Live View AF or MF). Useful for perched birds, not birds in flight!
With your current setup you could still go with an APSC body if the cost and weight aren't prohibitive. I'd go that route over the 5D MkIII and TC since you'll have better AF, more AF points and be at an effective 640mm f5.6. A little more noise maybe if you're shooting at high ISO but I think the images would likely be better than using the 1.4x on the 5D MkII, and with the 5D MkII you can still use the 1.4x TC and get center zone AF at an effective 896mm f8. If the weight is too much, you could always leave the 5D MkIII at home if you know you're mainly going to be doing long telephoto shots!
You can get a
7D MkII (open box) for $1250 ir a new one for $1500 or a
70D from $799 It's generally true that an APSC body will be better than an Full frame body + TC for long telephoto readch, assuming that low noise at higher ISO settings isn't the deciding factor.
So my kit for telephoto shots of small subjects would probably be the 7D MkII, a Sigma 150-600 - and a used MkI or Mkll 14x TC for when my subject was sitting still!
The one advantage of the Canon 100-400 MkII is that the AF may be faster and more accurate when tracking birds in flight than a 3rd party lens. But I'd still put it on a 7D MkII.
A lightweight Rebel body is also an option, but you're out of luck if you demand AF microadjust. The 70D has it, but none of the rebels be. The Sigam 150-600 has it though, via the Sigma USB Dock (see
http://bobatkins.com/photography/reviews/sigma_usb_dock.html)
There are so many options here, it's hard to pick one, but going back to my opening comment, with the gear you have, I'd go with a used Canon 1.4x TC and soo how that works out. It's light and it's relatively cheap. If it doesn't do the job, then you can consider other options.