As far as AF accuracy is concerned I'd say in general the canon red ring should be more accurate.
AF and capturing "childhood" is something I take seriously. I have absolutely zero tolerance of inconsistent AF. Between the two mentioned I'd take a 135L.
However on the other hand I have a preference to take in some of the environment in the frame. 85mm does give some story telling while paying great attention to the subject (kids/headshot/portaits). Only canon 85mm for my gearbag though....
If your 5dmk3 is somewhat newly acquired I'd revisit the 85 f/1.8. "good" colours with accurate fast AF and most of all extremely affordable. All fast primes have somewhat of purple fringing but the 85Lmk2 exhibits that PF aswell.
Bulk of my photos (I have 3 kids) wideangle is most preferred for beautiful documentation of growth. If you get "in your face" 135mm focal lengths I'd certainly make sure I supplement with wideangle.
135Lis the best bang for the buck redring along side your awesome 17-40L.
As far as colour is concerned I still think the 100L has better colours than the aging 135L.
For my run and gun kit at family outings I sometimes bring my 24mm and 85 f/1.8 as a light versatile prime combo. In close quarters or even family walks the 24mm with full frame allows me to frame a bit more environment. Looking through thousands of photos I really enjoy the wide angle lenses (16-35, 24mm) since it stirs up memories much better than headshot or a fame primarily full body. 135L would be used for very speicfic "looks" 35mm isn't all that special to me but I still keep my 35L around (gave the wife my 5dc with 35L combo).
If you want a blown out creamy bokeh the 135L would be great. For the sake of capturing the moment outdoors you already have one of the best portrait lenses from canon. Your 70-200 mk2 is killer awesome. I still think the 135L will separate you and your child in a longer distance so a shorter FL seems more suitable for a little runner.