Hawaiian songs about whaling exist in writings left by Hawaiian seamen, but no chanteys found so far. Some 5000 Hawaiian seamen are listed in port shipping records, 1842-1867 alone.
The newspapers of Hawai'i from 1834- may have poetry and songs, but they are only beginning to be digitized.

Kaulana ke anu i Alika,
Ka lalawe i ka ili a puni.
(Famous is the cold of the Arctic,
Overwhelming your entire body.)

The first account here has some interesting descriptions of the Inuit.
http://scholarspace.manoa.hawaii.edu/bitstream/10125/6778/1/JL40105.pdf

The following are the first few lines of a song by Ku, of O'ahu, a whaling song he wrote in the 19th c.

Makemake na Au e 'Ike ia Kaleponi

Makemake na au e 'ike ia Kaleponi
I ka 'aina o ka nani a me ka maika'i.
Maika'i 'oko'a no ke kai kuono o
Hukekona.
He nani Papine me Kaliona.
Ka home i aloha 'ia na na holokahiki
I aloha 'ia i ka leo ho'oholehole o ka
hulipahu,
I na olelo ke aloha a ke kolomeki.

The entire whaling song by Ku in "Na Mele Welo: Songs of Our Heritage," Bishop Museum Press, 1995. Reproduced in the following, p. 46. No translation provided.

http://pnclink.org/pnc2005/chi/Presentation-PDF/081-Susan%20A.%20Lebo-Aus%20Panel.pdf