Flash does NOT bother animals.
I use Flash to fill the shadows frequently in my own wildlife Images, in particular when I'm on Safari Shoots in Africa, in early dawn, late afternoon it can be an Image saver.
You are not restricted to 1/200th, at least not with the 1Dx & I think the 5DMKIII, check your Manual re High Speed Sync.
I agree with Sanj, it's been my experience to date that Animals aren't concerned about the Flash, of course that's an assumption based on results to date, it must be said I've never actually had one come up and tell me to "turn the bloody flash off, or else".
The attached Image is very difficult to have lit well if you don't use some Fill Flash, low light at Dawn, inside the tree canopy, this was shot with the 1Dx & 600 EX-RT off Camera with the ST-E3-RT.
Be careful of "Green Eye" with Predators, not the Envy thing, if you don't get the Flash off camera sufficiently, you have the light reflecting back from the Animals inner eye, very hard to get rid of in Post, distance from the Lens is the Key.
I do have the better beamer but have never tried it, some results from People look impressive, I can see for Birding Photographers this could be a real asset.