Just some thoughts …….
Well the ‘navy captain’ could be hearsay. At least you have documentation for ‘perfumer’ and now ‘constable’. As a town constable I wonder if the Winchester archives would have records? That could be followed up.
I am sure you would have explored the newspapers more using ‘constable’ in your search.
William was a “gentleman” when married – and his family clearly wanted to hold on to that grand title, even though many years later, he was not. Money does eventually run out, and William and his wife clearly had to earn a living.
To be a gentleman in one’s youth suggests he inherited money – directly from a parent, or from a grandparent, or uncle or aunt.
I think the Pinckney - Atkinson family of Amesbury should be explored further (See reply #18). It would take some time, so it really is up to how deep you want to delve / your level of dedication.

It would involve going a few generations back and working sideways through all the siblings.
There are some Wills for Pinckney and Atkinson family members at the National Archives – free downloads. With some reading, one might yield results. I would start with the ones from Amesbury.