A bit more info on the source of the attribution to Hamilton. They date to Charles Gray’s “Cursory Remarks on Scottish Song” in Glasgow Citizen, 1841.

George Farquhar Graham, Songs of Scotland:

Captain Charles Gray, R.M., in his “Cursory Remarks on Scottish Song,” says, that the believes “Burns did not write more than the first sixteen lines of this beautiful song.” He also observes that the third and fourth
stanzas were not found among Burns’s MSS. after his death; and that none of his editors or commentators, except Allan Cunninghham and Motherwell, have claimed them for Burns. Further, that Dr. Currie in his edition of Burns, Mr. Stenhouse in “Johnson’s Musical Museum,” and Mr. David Laing in his additional notes to that work, do not mention these stanzas as of Burns’ composition; and that Mr. George Thomson, in his “Melodies of Scotland,” (edition of 1838), has rejected them as spurious. By some they have been ascribed to “William Reid, Bookseller, Glasgow; but Captain Gray is rather inclined to believe they were written by John Hamilton, Musicseller, Edinburgh.

This is pure conjecture on Gray's part, now reported as fact. There are many Burns songs and poems with missing MSS. Again, there is no real world motivation for Thomas Stewart to solicit fraud texts when he had proven access to manuscripts through his uncle John Richmond. -Andrew Calhoun