**I had composed this and ready to post when Neale posted and they overlap somewhat with train of thought.
I think you need to look at Mary Elizabeth Sayles in records.
If she is the one who married Mr Harmer, her father was Edwin.
That family are at 1881 651 /56/31
Mary Elizabeth is born Westminster and they are living in Battersea.
There doesn’t seem to be a sister Mabel.
1921 Mary Elizabeth is born Clapham, which appears to be close to Battersea.
1911 gives details of several Harmer children - DNA might be useful here.
I might have missed this but are the notes on the death certificate not shown in full at all?
Whoever is the mother of the young Joseph Brooks does not seem to have any responsibility for him so you would think they would not be interested in claiming any pension on his behalf.