Could you put a white piece of fabric on top of the fabric you are embroidering on? If the breed of dog is white, then this would show as "fur" when there aren't any stitches on it? Kind of like an applique thing? Depending on how dark the base fabric is and how light your applique fabric is, you may need to add a second layer, or a very light weight interfacing.
Also, you'd have to trim around the outer edges as well....