There is a very interesting entry in the French Hospital London records
They had an application from one Thomas Spencer in February 1906. He was a Silk Weaver, born in Bethnal Green at 11 Squirries Street on Nov 9 1832 (baptised at St Leonards Shoreditch 9 Dec 1832) to Thomas Spencer and Susannah Pashley. He claimed his right to reside in the Hospital through his mothers French Huguenot descent Pashley/Cordon. Both his mother and father were born in the East End. His father Thomas Spencer was born 1802, to Thomas and Mary Spencer, and baptised at St Leonards Shoreditch on 29 Sep 1802.
More importantly he provided both his parents marriage certificate and his paternal grandparents.
20 Feb 1798 Thomas Spencer & Mary Shelton, St. James' Westminster
2nd Jany 1832 Thomas Spencer & Susannah Pashley, St. James, Westminster
So despite both families living in Shoreditch, as silk weavers, they were trapsing over to St James in the West of the city to marry. The reason for this I am not sure, though there surely must have been one.
Is it not possible this mans father was a brother to your Charlotte?
I think this is clear evidence the Spencers though were English and not French, or he would have claimed decent from his father not his mother.
I think there is a strong case to be made that the Thomas Spencer who married Mary Shelton, 1798, is a brother of the David Spencer I researched and also originally of Essex. Not only are they living in same parish following same trade, but the petition to the French hospital also mentions the applicants own wife, an Eliza Diggins. David Spencers wife Mary Spiers, had a sister Lydia Spiers. Her son James Houghton married into the Diggins family too. (The Houghtons and Diggins were both also native English Silk Weaving families based in Shoreditch back to at least early 1700's)