Not everything can be fixed in post, more so with compressed video formats. Especially blown highlights. Once they're gone, you can't get them back. Shooing RAW stills offers a degree of control you just can't get unless you're also shooting RAW video.
In short, your best option is to fix this on set, not in post.
As a colorist, I deal with these issues a lot. You likely got better results in Lightroom because the pictures you were grading/correction were in a more workable compression format/color space.
What I would recommend is taking the footage into a program with a color correction application that supports secondary adjustments (After Effect's Synthetic Aperture and Magic Bullet's Colorista both support secondaries).
You can then use the secondary adjustments to isolate the problematic areas and then correct them. The one disclaimer is that you can only correct as far as your compressed codec will let you (unless you are using a RAW camera like a RED or an Alexa).
Hope this helps!