The tune intended can in this case reasonably be deduced from the text. The late Bruce Olson had this to say in his study of the 'Granuaile' tune and various songs associated with it:

'For another historical song obviously to this tune, and in which Granu Weale reappears, see "The Hornet and the Peacock" in the Digital Tradition Database. Thanks to Dick Greenhaus for this reference. A fragment of it (which does not mention Granuaile), and which is to a different tune is "The peacock that lived in the land of King George" in Mary Eddy's 'Songs and Ballads of Ohio', #107.'

http://www.csufresno.edu/folklore/Olson/SONGTXT1.HTM#GRNWALE

In fact, 'Granuwail' as it appeared in B Cooke's Selection of Twenty One Favourite Original Irish Airs (c1795), does bear a noticeable resemblance to 'The Laird of Cockpen' (formerly known as 'When She Cam Ben She Bobbit'). Ives doesn't say where he got his text or tune, but the latter appears to be a form of 'Granuwail'.

The second broadside at American Memory is a quite different song on the same subject, and intended for a different tune.

The 'Hornet & Peacock' broadside carries the imprint 'L. Deming, No 61, Hanover Street, Boston, and at Middlebury, Vt'.

The bookseller and barber Leonard Deming (1787-1853) seems to have operated from 61 Hanover Street from around 1837, having previously been next door at number 62; and prior to 1832, at 1 Market Square Corner. That's rather after the actual battle, so presumably there were earlier editions. (Details extracted from http://hubpages.com/hub/American-Book-Trade-Index).