Looks like Burns probably did his usual with this one and altered an existing song ... at

least BruceO's entry for Scottish Tunes href=http://users.erols.com/olsonw/SCOTTUNS.HTM>here would suggest this:

To Daunton me [SMM #182. Glen, ESM p. 120, cites tune in Atkinson MS, 1694, as "This Wife of mine."]/; Be Valiant still [Logan, The Pedlar's Pack, p. 164, 1869, notices this as the title of a broadside ballad of c 1700, sung to the tune of "The old carle to daunton me." A song of this title, and to the tune of "To Daunton me," given in J. Hogg's Jacobite Relics, II p. 89, is evidently of much later date.]; MHF 38: To Dauntin me; OS1 38: To daunton me; MG2 27: To dauntin me; CPC1 16: To danton me; MBG2 23:

The phrase itself also occurs in "Thomas The Rhymer" (Child, Part II., p. 317. usually attributed as C17th), which Burns knew well.

"Harp and carp, Thomas," she said,
"Harp and carp, along wi' me,
And if ye dare to kiss my lips,
Sure of your bodie I will be!"

"Betide me weal, betide me woe,
That weird sall never daunton me;
Syne he has kissed her rosy lips,
All underneath the Eildon Tree.

The word Daunten is quite popular in Middle English, having been used by Langland (Piers Plowman) "And David shal be diademed and daunten hem alle" as well as Chaucer (Romaunt of The Rose) "The god of Love, that can devyde/Love, as him lyketh it to be./But he can cherles daunten, he,/And maken folkes pryde fallen."

:-)
Ian