Use Keylight. It comes with After effects. For screen color choose the white background (as you are probably aware). Use the alpha bias and select the dog's fur. Then go to view and select "Combined Matte." It will change to a white/black/grayscale. Next go to screen matte, and play with the clip black and clip white, and the despot black/white options. If you are still not happy with the key, go to the screen color, click on the swatch to open the color menu, and then vary slowly drag the color lighter or darker to see if that helps get rid of the remaining issues.
First, shoot the photographer for thinking this would be an acceptable shot. You're now in charge of damage control.
Second, if Derek's Keylight suggestion doesn't work well, you may have to use Rotobrush. If you have no experience with it, you'll need to read instructions and run the many tutorials on it. It is not at all intuitive to use.
Third, I'd be tempted to sidestep isolating the dog from the shot. I'd rather try matching the background to eliminate the offending junk in the shot.
And if all else fails, just put a mask around Fido and put the shot over an altogether different background.
KGAN (CBS) & KFXA (Fox) Cedar Rapids, IA