There are two copies of Barnaby Fin[n]egan at  Bodleian Library Broadside Ballads.  The following is the more legible, and is significantly different from the version quoted above:

Barnaby Finnegan  J. Scott of Pittenweem and J.Wood of Edinburgh.  No tune is specified.  There is a tune in O'Neill, however, which might be a possibility; it can be found via  J C's Tunefinder:

Barney O'Finnegan.  The coincidence of title may of course mean nothing, but I suppose it's a start.