The title of the song says that the Dumbarton Castle plied between Glasgow and Inverary at the head of Loch Fyne¹ (see map). 

She went to the Forth in 1819 but later returned to the Clyde. She appears to have been sunk off the Irish coast c1838

Reference:
An article by Graham Lappin published in "Clyde Steamers" magazine and given to the curator by the author