I did not hear about this condition until yesterday, so I googled "red mange" and found some interesting info,; specifically a "mite" called Demodex Mites burrow in the hair follicles and skin of certain species of dogs. To your specific question about using zinc oxide, I do not know about the remedies except what is discussed on several of the websites. But you may check to see if there is any research (vets or universities) using zinc oxide against these pests. According to what I've read, there are several treatments (Amitraz, Tactik, Mitoban or Milbemycin Oxime) that help along with weekly dippings.
Hope some of this helps.