100-400L and a 50mm are all that I have ever needed when going to the zoo.
400mm is definitely needed. 200mm is way too short. Using an teleconverter with your 70-200 would be the cost effective way.
I find carrying around a lot of gear awkward depending on the amount of people at the zoo. Bad enough just walking around with a DSLR and a big lens. I hit a kid on the head with the lens hood when I was swinging the camera up to use. Not hard though. Changing lenses is a bit of a pain in the back side. That is why the 100-400L was so appealing to me. The Sigma 120-400mm f/4.5-5.6 DG HSM OS APO is a great lens too. A little softer at f/5.6 on the long end compared to the Canon L. A lot cheaper though. Stop it down to f/8 and you are flying. If you get one used just for the trip you can always sell it again. Or rent either of the lenses. The Sigma 150-500mm did not appeal to me. Not much difference from 400mm to 500mm. But the maximum aperture drops to f/6.3 around 300mm. Only 1/3 stop, but you have to shoot at f/8 already to improve on sharpness.
At 400mm you can easily shoot through the cages without seeing the bars and blur out the background to hide any signs of the hand of man (fence, cage, buildings, etc). You get more frame filling shots.
Check out my website. Go to the animals gallery to see images form the zoo with the 100-400L. On my first visit I brought four lenses. I was constantly changing between a 70-200 and a 400 prime. On visits II to IV I only brought the 100-400L (I sold the 400 prime) and either a 50 or 85mm prime.
I'm in Canada. Isn't that weird!