My wife watched this several times, wondering the same thing. She finally came up with the theory that the person who murdered Archer Coe also killed the dog. This person (revealed at the end of the film) is seen at the Kennel Club after the dog is killed. The motive for killing the dog would be to implicate another suspect by providing that suspect a motive for killing Coe.
Philo Vance, Hilda Lake, Sgt. Ernest Heath, Dist. Atty. John F.X. Markham, Archer Coe, Brisbane Coe, Dr. Doremus, Sir Thomas MacDonald, Gamble, Doris Delafield, and others
What is the plot of The Kennel Murder Case?
Philo Vance, accompanied by his prize-losing Scottish terrier, investigates the locked-room murder of a prominent and much-hated collector whose broken Chinese vase provides an important clue.