Hello Rob Dang
I'm not sure if this this is the same Hamilton family, but I have a Constantinople-Hamilton connection too.
My ancestor, Ariston Vincent CASTELLI, born 9 Apr 1830 in Constantinople, married Charlotte 'Lotti' HAMILTON (1852-1940) in Camberwell, London in 1874. Even though they married in England, they lived in Constantinople as a married couple because all 3 of their children were born & baptised there. Castelli died in Constantinople and was buried there in 1906.
Lotti was born in Liverpool, 1852, daughter of Henry HAMILTON (d. before Jan 1881) and Eliza (b. circa 1830). The death of Henry was recorded with the Italian Consulate of Constantinople because his son-in-law Castelli was an Italian citizen of Constantinople.
Inscrite le 29.1.1881 au Consulat d’Italie de Constantinople - D 359
Carlotta fu Enrico Hamilton nata a Londra nel 1852
It could be a coincidence, but how many Hamiltons were there in Constantinople in the 1860s? It's very possible your gg gf Charles Hamilton of Shoreham and Lotti's father, Henry Hamilton of Liverpool were related, or at least knew each other. All the Europeans in 19th Century Constantinople moved in the same social circles and lived in the same neighbourhood called Pera across the Golden Horn strait, where all the Christian churches were. 'Pera' means 'Over There' in Greek. That quarter later became known as Galata, now called Beyoğlu.
There's a good chance your gg gf Charles married Consolata Caruana in one of the churches in Pera. If Consolata was Maltese, she was most probably Catholic.
I'll have to look into it a bit further.
Sources: https://gw.geneanet.org/marmara2?lang=en&pz=mathilde&nz=d+andria&p=ariston+vincent&n=castelli