I used Tinkercad to fill-in the Bernese mountain dog from Xilophone's wall decoration and increased its size by about 1/16". I then created the back plate and used the filled-in Bernese dog to create a negative space.
After printing the inlay and backplate I simply used a rubber mallet to force the dog's face into the negative space. It was really tight.
I designed it so the dog's head would be raised above the back plate, providing some depth to the hanger.