It seems there are two basic manifestations of this ballad. The one we generally hear seems to originate in a broadside that may have been penned and circulated by Lady Franklin herself. These are the "Sailor's Dream" type. There is one in a different meter that describes the voyage in detail, including encounters with the doubtful esquimaux and lines like

"What hope can scale this icy wall,
High over the main flag staff?
Above the ridges wolf and bear
Look down with patient, settled stare
Look down on us and laugh"

"A Ballad of Sir John Franklin," from  Sartain's Magazine, May 1850
by George Boker
http://www.ric.edu/faculty/rpotter/sartain.html

Another source of odes was a poetry competition sponsored by the British Association which yielded a variety of epics.
en.citizendium.org/wiki/John_Franklin

Among the entries was this by Algernon Swinburne.Though it did not win, it is generally thought to be the best of the lot by today's standards
http://www.ric.edu/faculty/rpotter/swin.html

Another is "The Life The Character and The Death of Sir John Franklin" by Edward Owen. Typically Victorian, these, needless to say, have never become songs that I know of.