As best as I can tell, all sources point back to David Hume's (sometimes credited as David Hume Godscroft) The History of the House and Race of Douglas and Angus, pp. 143-144 (link) or the 1644 edition (link), p. 77. Both present the same information regarding this song. To spare everyone else from having to cipher out the old long esses, I've reproduced it below, skipping over some of the expository information matching that of the Ballad Index:

The Lord of Liddisdale being at his Pastime, hunting in Etrick-Forest, is beset by William Earl of Douglas, and such as he had ordained for that purpose, and there assailed and wounded and slain beside Galsewood [Galeswood] in the Year 1353, upon a Jealousy that the Earl had conceived of him with his Lady, as the report goeth, for so says the old Song:

The Countess of Douglas out of her Bour she came,
And loudly there that she did call,
It is for the Lord of Liddisdale
That I let all these Tears down fall.

The Song also declareth how she did write her Love Letters to Liddisdale, to disswade him from that Hunting. It tells likewise the Manner of the taking of his Men, and his own killing at Galsewood [Galeswood], and how he was carried the first Night of Lindin Kirk, a Mile from Selkirk, and was buried within the Abbacy of Melross [Melrose].

In addition to Hume's text, Francis Child also points to Walter Scott's Minstrelsy of the Scottish Border, Vol. 1 (link), which itself sites Hume (as Godscroft) in footnote 61, referenced on page 125. Perhaps a modern reader can, with the help of advanced searching, find the "ensuing" references? (vol. 2; vol. 3). There were a few that caught my eye but I'm sure someone much more knowledgeable than me could say with more authority.

The 1714 text Scottish Songs. In Two Volumes similarly points to Hume / Godsworth as the source, but notes that "This song, if extant, must be a prodigious curiosity."   Fancis Gummere (The Popular Ballad - 1907) surmised that "this more serious case of marital troubles seems not to be true," likely echoing comments made in Blackwood's Magazine (March 1887, p. 342) which noted that "there was no Countess of Douglas at that time," and goes on to cite Fraser, who believed the actual cause of death was over a dispute of hunting lands; a similar viewpoint was proferred in Scottish Songs. In Two Volumes