I'd guess the above took place in nice warm waters, none of your Greenland fisheries there. He'd have been frozen ta death in a minute, otherwise.
I'll have to play my video of Moby Dick again to make sure, but I recall no less than three different versions of Paddy Doyle being sung, one while stowing a bunt (heaving up the furled sail to the spar) one as they flayed the whale, and another as they were rowing back to ship, hauling a dead'n. And who said A'Roving aint a shanty? In the film it was sung (and danced to) in the pub when Ishmael met the crew, but Hugill (may his name be revered) gives it as a capstan song, also used at brake windlass and pumps. Then again, you could sing any bloody thing at pumps if the damned job was long enough!