I have had a lot of success recently in tracing back the Earnshaw line in Yorkshire. This includes finding that my gg grandfather was baptised by Patrick Bronte. Also found many additions to the tree.
But my reason for responding is that you said you were interested in Brighton. I had a relation who was born in Brighton (Nellie Sunderland Earnshaw), but then found a William S K Earnshaw who was also born in Brighton. I purchased the birth certificate, but found it wasn't connected as far as I can tell. I wondered if this is one of your family.
The birth was of William Stanley Keyworth Earnshaw, son of William Thomas and Helen Earnshaw. This was in June 1902. Is it of any interest to you?