Although I have nothing directly citing Hawaiian chanteymen, there are many references to Hawaiians in the service of the fur trade, whalers and merchantmen, California hide trade, HB farms and HB construction, etc. These random notes may put some perspective on the period.
An important article on the subject is Quimby, George I., 1972, Jour. Pacific History, vol. 7, "Hawaiians in the fur trade of north-west America, 1785-1820."
Sir George Simpson, Governor-in Chief of the Hudsons Bay Company, 1847, "Narrative of a Journey Round the World during the years 1841 and 1842" wrote as follows about his trip in a boat along the Columbia River: "Our bateau carried as curious a muster of races and languages as perhaps has ever been congregated within the same compass in any part of the world. Our crew of ten men contained (an) Iroquois..., a Cree half-breed of native origin, a north Briton who understands only the Gaelic of his native hills, a Canadian... who knew French; and Sandwich Islanders who jabbered a medley of Chinook, English etc., and their own vernacular jargon...; add to this (passengers natives of the British Isles, Russia etc.) and you have the prettiest congress of nations..."
The Hudsons Bay Company records speak of the Kanaka voyageurs, whom they found more reliable than many of the French, and as good as the Metís.
Sir George speaks of the whaling ships, "Whaling ships have left San Francisco" .... have all gradually betaken themselves to the sandwich Islands." ... "The Sandwich Islands afford to the refitting whaler an ample supply of competent labour, both native and foreign, at reasonable wages." (He spoke of the population of San Francisco as "naturally indolent."

After the abandonment of the feudal system, many young Hawaiian men were left without occupation. For a short time, until joined by both American and English advisors, the "missionaries notoriously became, so far as new legislation was concerned, the real rulers of the Group." King Kamehameha II spoke English and French fluently
The Royal government, with the help of these advisors, arranged contracts for the placement of Hawaiians in employment in sea-going trades, the fur trade, sawmills, construction and farms in western America. Contracts were for a set period, and the employers were required to provide for return of the Hawaiians at the end of the contract period. In the Hawaiian Archives in Honolulu are many papers relating to these contracts. One discussed in Repts. Hawaiian Hist. Soc. was an agreement between Kekuanaoa and George Pelly (1840) to take 60 Hawaiians for service in the Columbia River, for a period of three years, "to be returned at the end of said period term on penalty of $20. each."

Fort Langley in British Columbia was constructed with Kanaka labor; they were the sawyers. The first HB sawmill, 1828, five miles upstream from Ft. Vancouver, was operated by Kanakas. By 1830, 200,000 board feet of lumber was being sent to the Islands. Salmon was salted and barrelled by Indian and Kanaka laborers, timber was cut and finished into lumber for shipment to the Hudsons Bay Store in Honolulu, etc. The farms experimented with various crops.

After the merger of HB and NW Cos. in 1821, the Kanaka labor force was a key factor in HB operations. The Indians were 'unreliable,' and French Canadian voyageurs were expensive and 'independent'. Chief factor at Ft. Vancouver, Dr. John McLaughlin, offered 10 pounds/year to islanders; by 1823, some 200 islanders were trained sailors. In addition to working as seamen on coast vessels, they built boats and worked on the canoes and York boats, and in various positions at the Forts.
Kanakas were important before the HB period; the crew of the Astor ship, 'Beaver', in 1812 consisted of five Canadians, seven Americans and twelve Kanakas.
At Fort Walla Walla, 1818, 25 Canadians, 32 Kanaka and 38 Iroquois were employed under Donald Mackenzie. The Iroquois attacked one night, but Mackenzie was saved by the Canadians and Owhyees.

The Hawaiians' 'warrior ability' also was put to use by HB to assist in pacification of local Indian tribes- the first punitive expedition was led by A. McLeod in 1828 against the Clallum Indians. In addition to the 'Owhyees', Chinooks and Iroquois were in the force.

Simpson spoke of Hawaiian women riding horses "en cavilier" on the Spanish saddle; Mexican vaqueros had trained Hawaiian cowboys ('paniolos') beginning in the 1820's after cattle introduced by Vancouver needed control, and cattle sales became important.
An important part of Hawaiian music is the result of the mixture of Hawaiians and Mexicans in the cattle trade, later reinforced by Canary Islanders, Portuguese and others.
Kanakas were employed by the Spanish in California in the hide trade (described by Dana in "Two Years Before the Mast").

In 1849, the Advertiser (Honolulu) reported that Rev. S. C. Damon visited Kanaka diggings in California, where 75 Sandwich Islanders were searching for gold.

Some of the Hawaiians stayed, and small settlements developed in British Columbia, Washington State and elsewhere. There are articles about Hawaiians who settled in New England when they were a part of the sea-faring trade.
Not all stories had happy endings; in 1903 an article in the Rept. Minister of Foreign Affairs describes the plight of indigent Hawaiians in Utah.