There are several things you can do to that picture. It would be best of course to have the raw files. Then you should be able to make reasonable adjustments but still use the full dynamic range and resolution of the output format. I used my own software for this, but these are all ordinary operations any photo post-processing software should be able to do.

Here is what we are starting with:

First, I let the software find the darkest and brightest points and set them to black and white, respectively. Actually to be precise, it finds the dakest and lightest values in every block of 4x4 pixels average.

You were right, the brightest spot was definitely blueish. The darkest is (.031, .058, .094) and the brightest (.878 .916, .955). The snow in the foreground isn't the brightest spot, a cloud at top is. Clouds are not good for color ballance, so we'll try to make sure the snow is white, or actually a shade of gray near white. If we made the foreground snow full white, the highlights would be lost at the top. In some cases that's OK, but let's preserve them in this case.

A reasonable size rectangle in the untouched forground snow has the average (.793, .840, .895). If we make that 95% white, then there is still a little room for the brighter spots near the top. The important point though is that we're taking the color ballance from the foreground snow by declaring it a shade of gray.

The color ballance is now solved, but I see something else I don't like. Note how the near trees look almost totally black. There was some detail visible there before, but only because you essentially had a gray offset added to everything. A better way to fix this is to leave the darkest spot black, but bring the intensity up more quickly at the low end. In this case I used my brighten adjustment with a factor of 0.8. I am sure other software has similar things, but probably called something a bit different.

Note how there is now some detail in the near trees but the darkest spots are still black. A side effect of this is to make the trees in the center look more distant. This is because they are farther away and more haze was between them and the camera. This added a black offset to them. The brighten adjustment was applied globally. It lifts the brightness of near-dark things, which these trees were. The haze was real and this picture is showing it. Only you can decide whether that's what you want though. If not, you'll need to to some selective masking. Global adjustments don't work well on haze. Haze also tends to be blueish. If it were me, I'd probably go for having it make the depth of the picture more apparent, but this is getting into asthetics and no longer a technical adjustment like fixing the color ballance.