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.
Best,
Derek Boettcher
Creative Management
BTH Media Group
https://bthmedia.com