That does sound like you have done a lot of work already. If there is only one likely candidate it does sound promising.
Louisa had 4 children, did she marry? Do you have a marriage certificate? What age does it have for Louisa - at the birth of her children and at her death, presumably these give you the indication she was born in the 1860s, but is it more specific? 
You've probably seen the passenger list on "The West Australian" has her name Louisa STAMPS, but I've looked at the actual image and it definitively STAMP.
https://trove.nla.gov.au/newspaper/article/2991651
As far as finding the records for Felgates, I don't have any better advice than was given previously, maybe contact the National Archives, or the London Metropolitan Archives.
Will keep looking ...
Hi Maddy
Thanks for your continued support. She met a cad, John Carter (my 2x GGfather) not long after arriving in Australia. He had had 4 children with his wife Sarah Saxby in Sydney before abandoning them all and moving to Perth. He met Louisa and had 2 children with her, then abandoned her.
She tried to get the two children into the poorhouse here, so she could get work to support herself, but was unsuccessful. The police officer reporting on the situation said "she is a quiet and sober woman...she does not appear to have any associates and is quite friendless, the people from whom she rents her room will give her a few more days grace, after which she will be turned adrift with her two children." Breaks my heart every time I read that
They later reconciled and had two more children.
Her death cert says she was 55 when she died in 1916, making her birth around 1861.The informant was an assistant undertaker. Only two of her children's birth certs have her age...27 in 1896 and 30 in 1899, making her birth around 1869. She was the informant for these birth certs. It seems unlikely she migrated to Australia by herself when she was only 15, so the truth is probably somewhere in between 1861 and 1869. (The Louisa I found in the UK censuses was born in 1867).
Yes I saw the passenger list, and you're right...I think the journalist writing the entry mistook the big 'p' at the end of 'Stamp' for a 'ps'. I also tried researching the other passengers, as there's only a few. I thought she may have migrated with friends. No luck
Though the other lady called 'Catherine Ringrose' in the newspaper looks to have a different surname on the passenger list. Can you make it out at all?
Cheers
Rob.
