Joanna Colcord "Songs of American Sailormen" has (p 154) "Franklin's Crew" but this is the common one put partly in Lady Franklin's mouth. So far as I can see there is no version of any song about Franklin in Gale Huntington's "Songs the Whalemen Sang"

There are other songs about Franklin and I think I copied a ballad sheet recently but haven't time to look nat the moment.