Butternut wrote several items.

"Cape Cod Girls" with that approximate form appears to go back only to Joanna Colcord (1924), but it is intimately linked to "South Australia," which is a few years older.

"Two Dead Boys" in that form doesn't seem to have seen print before the late 1940s (Opie & Opie, I Saw Esau, 1947; Withers, Rocket In My Pocket, 1948).

Your citation of Blind Pig and the Acorn is reading more into their statements than is actually there, I think. Yes, "The Land of Cockayne" is a tale of things impossible, but it is clearly not related to "Two Dead Boys." If you want to see the text, the authoritative source is Thorlac Turville-Petre, Poems from BL MS Harley 913: 'The Kildare Manuscript', Early English Text Society/Oxford University Press, 2015. It has both text and commentary, though I doubt the text will be much help; it begins

Fur in see bi west Spayngne
Is a land ihote Cokaygne,
(Th)er nis lond vnder heuenriche <-- That's a thorn, not a Th!
Of wel, of godnis, hit iliche.

Which I would paraphrase

Far at sea, to the west of Spain
Is a land called Cockaigne
This is no land under heaven
Of good, of godliness, like it.

The 1480 citation, of Bodleian Eng. Poet. e.1 (Bodleian 29734) is a much more interesting source; it's a "Small manuscript, 4.375x6". 65 folios (although, because two folio are numbered 27, the last folio number is 64); the first 10 folios were added in binding. Two scribes were involved, with the first writing most of folios 11-50 and the second writing the rest and making occasional changes to what was written by scribe A. Three pieces have music. Based on the count in the Brown/Robbins Index of Middle English Verse and its supplement, it contains some 75 Middle English poems plus a few Latin pieces." It was printed by Thomas Wright, Songs and Carols, Percy Society Publications (a book I haven't seen; copies are obscenely expensive). Unlike the Kildare Manuscript, it has several pieces that were pretty clearly traditional, or at least popular: "Bring Us Good Ale," a "Holly and the Ivy" version, "O Mary Mother," "This Endris Night." I considered a dozen and a half pieces in it to be folky enough to put in the Ballad Index. But I haven't seen anything in it that I'd consider ancestral to "Two Dead Boys." Again, it's just the style of a few items -- the "things impossible" type.