That's pretty good quality. I think you're doing well. The other thing I did was practice different metering and see how it affects the exposure. For instance, you could do spot metering on the bright side of the bird, take the reading, and adjust the shutter speed accordingly for proper exposure. Usually, when I do something like that, I'll take 3-5 photos with AEB, -2/3, -1/3, proper, +1/3, +2/3, and then do two metering sites. You have 10 photos then, and one will certainly be the way you personally want it. That's just what I do. I know it can add a lot of RAW pictures that you're not going to use and have to sift through, but in this case it might be worth trying once or twice and seeing what your options look like. And it's fun!
However, in this case you may not have had time to do that, as the bird probably took off within seconds, maybe. I think the 5D Mark III can do spot metering everywhere (all 61 points) in single shot mode can't it? I can't remember. AI Servo is different.