I am also a descendant of Edward Pickering (1839-1906), twice mayor of Durban.
The misinformation that your grandmother was his daughter is likely due to the fact that he did have a daughter named "Edith Mercy Pickering." However, she was born on 1 August 1872 "at the Berea" in South Africa. She married William Moyse Henderson.
If your Edith Pickering was a descendant of Edward Pickering (1839-1906) she must have been a daughter or granddaughter of one of his sons. I do not have all of Edward's grandchildren and great grandchildren in my files, however, I am interested in these lines of descent. Edward married Harriet Camp on 1 July 1859 in Brighton, England. They soon sailed for Durban where all his children were born. There he practiced his trade as a cooper, then merchant, and finally owner of a department store on what became known as Pickering Street.
His 5 known sons, identified in their father's will and a newspaper death notice, were:
1) Benjamin Alfred Pickering
2) Edward Henry Pickering
3) John Ethelbert Pickering
4) Joseph Thomas Pickering
5) Albert Edgar Pickering (b. Jun 1864, d. 15 Jan 1876)
I have no information on the other sons, but my great grandfather Joseph Thomas Pickering married twice, first to Blanche Gertrude Pickering and, after her death in 1905, next to Kathleen (birth surname unknown, first husband's surname was "Mitchell"). My grandmother Blanche Gladys lived in Canada and the US. Kathleen had two children with Joseph, Fred born 5 Mar 1911 and daughter (?) Pat born about 1912. Kathleen was a widow with two daughters, Ivy Mitchell and Kathleen Mitchell. So Joseph Thomas Pickering is not your ancestor.
One minor difficulty is that the death notice for Albert identifies him as the sixth son of Edward Pickering. The one stillborn child known was identified as a daughter. This may have been an error or acknowledgement of another minor son's death.
If you want copies of printed information about Edward, I have a full transcriptions of his will and his death notice, as well as a pdf of a biography (and photograph) published in a Milwaukee Temperance magazine in 1897. Just ask.