Are you sure about the year and /or place the only children of William and Charlotte STUART, baptised in Harwich between 1813 and 1827 were:-
George bn. 22 Aug. 1822, bp. 23 Aug. 1822
Elizabeth bn. 4 Aug. 1824, bp. 5 Oct 1825.
There were six other STUART children baptised during that period, to parents Henry and Sarah. William and Hannah, Richard and Hannah, none of them was a William.
I have even looked at Dovercourt PR's just in case it was there, as my family moved between the parishes.
Jebber