I can't give an After-Effects specific answer, but maybe this will help..
1 - If you want to color a specific part of the image, deal with "region based filtering." In DaVinci-speak, that's Power Windows. Masks, Shapes, Roto. However you phrase it. That's how you tell the computer what part of the image you want to effect. Tie the mask to a tracker so you don't have to hand-animate it. Also, make the edges of the mask VERY soft. Don't try to mask the building exactly. Have a several hundred pixel soft edge on the mask, and even if the tracking isn't quite perfect, you can still get away with murder.
2 - A Local Contrast Filter. AKA "Fake HDR Look." Used sparingly, this sort of thing can be used without turning the whole image into a horrible fairy dreamland nonsense picture. It will help "pop" some of the low contrast regions without doing global color shifts.
3 - You may be able to gamma up the mids while leaving the lows dark. This would reduce the contrast between the lit and shadowed snow, but leave the dark rocks. If you have a really specific goal for what it is supposed to look like, it just won't be a 1-button process.
this video is amateur, do you mean that the footage is very shaky?