Dog in Bowlera Cross Stitch Kit by Lanarte

I am a senior stitcher with lots of experience and am eager to make this for my Canadian neighbor. Her little dog is a mix, but mostly dachshund. He had a longer coat, almost like a Cairn Terrier. How hard do you think it would be to add a sort of furry overcoat with more floss? I can change the color to match his terrier coat more, but adding rough long fur migt be hard on this little 3.5" x 3.5" piece. How would you aporoach this. His name is Cooper. His fur you can look up at the AKC site under Cairn Terrier or just think of the Toto dog in the 1937 movie of The Wizard of Oz!