From the CON31 record for Charles KING, "Bardaster" 1835, Charles is single.
Many years later, and Charles KING is marrying Catherine COUGHLAN, a woman at the end of her childbearing years. Is this a first marriage for Charles?
They have a son, 1851.......and this boy is named ......Thomas.
You might expect that Charles would give his own name to this son.
I would look for an earlier (unregistered) birth to this couple, and possibly found as a baptism record.
Accounting for all children born to Catherine in Tasmania holds the possibility of finding a record where Catherine's family name is recorded. If Catherine had registered the 1851 birth you might have seen her family name. Unfortunately, husband Charles is the informant.
Further to the purpose of finding any other child born to Catherine, can you identify the marriage and children born to Thomas KING please.
Grangegorman Prison, Dublin (Findmypast)
COUGHLAN Catherine, 38, received 25 Nov 1847
Children.....Ellen COUGHLAN....age......years...11....months...........
Ellen would be with mother Catherine in prison. Between Nov 1847, and ship departure, daughter Ellen has....what?.....stayed in Ireland.......died??......