Oh my goodness you are right! The informant was not William, it was Mary Anne Matilda. And the child was born at a house, 83 Stanley St, which is where the information was given to the registrar.
My own speculation covers similar ground to what you have suggested. I have searched in Trove for references to Edward Charles Pye but at this stage no luck.
There are many possible leads to follow but thank you for taking time on your Friday evening to help!