What about his clothes?

Guess you could not show a naked man back in he 80s. Maybe the are like furr or feathers and just belong to his body so they reappear even though you can see them rip during the transformation.

Well he is using magic to turn himself into animals, maybe could magically make his clothes appear and disappear too. If Hulk's pants can stay on, so can his clothes disappear and reappear :P