Here is an approach using a curve adjustment that mimics the process used in white balancing raw files (scaling the exposure of the R, G and B channels). You make straight-line adjustments to the R, G and B curves either by dragging the white point (the endpoint of the curve on the white end) to the left to increase the channel's scaling or downward to decrease the scaling. You need to find a reasonable neutral in your image and then see how the pixels in that neutral patch are represented in the R, G and B channels - if the patch is neutral, the pixels in the R, G and B channels should all have similar values and the histogram should show that the peaks in the R, G and B channels are aligned with one another. If they are not then you use the curve adjustment to get them in line. See screenshots.
Love the shot, btw.
Here is the neutral patch (blaze on your pup's chest) selected and the histogram of those pixels - if the blaze is whitish, the peaks should all be aligned, but obviously they are not!
So, we make some curves adjustments - here I reduced the scaling on the R channel and increased it on the B channel to bring the three peaks into alignment. Here is what the curves look like:
and here's what the histogram of the patch on the blaze looks like after the adjustment:
Here is the balanced image (I went toward red), with some additional tweaking of contrast. - you may decide that you want more warmth, etc., which you can do by making tweaks to the curve white point away from neutral.
Notice that I brought the reds down and the blues up to meet the green channel - this keeps the top of the head from blowing. What is apparent when you balance for the overwhelming tungsten cast is that there is daylight coming from a window illuminating some of the subject - the more obvious areas appear as a bluish cast (dog's nose highlight). Trying to balance for mixed lighting is a little bit more difficult, but you can just do the same thing on another curves layer and paint in the daylight-balanced curve adjustment in the spots where you get an intensified bluish cast from the first balancing curve.