The song was probably "The Gallowa Hills" and the tune was "Campbell's Farewell to Redcastle", a mid-19th-century Scottish pipe tune that got popularized in the US under several other names in the late 19th century (Nigel Gatherer knows exactly how that happened but I haven't heard the story).

"Believe Me if All Those Endearing Young Charms" uses an English tune from about 1700, "My Lodging is on the Cold Ground", and it isn't really all that similar to "Campbell's Farewell to Redcastle". The words for "The Gallowa Hills" also predate "Campbell's Farewell to Redcastle" by decades. So I think Archie got the chronology somewhat discombobulated.