I never heard Belle sing this, but there is a recording of it, by Jimmy Scott, on the Folkways album "The Borders" FW8776. His tune is indeed "The Day we Went to Rothsey,O". Your half verse above is not in Jimmy's version, but is obviously a varient of the same song.
From Samuel B. Charters' notes - "This very popular bothie song is also found in a similar version in Ford's Vagabond Songs. Ford states that it was writtenby William Watt of Peebleshire and first published in 1835. It is found everywhere in Scotland."
As I believe Folkways have kept all their releases available, on tape at least, you should be able to get a copy from them.
love, john.