'St. Lawrence Bay' of the song could be one of the bays of St. Lawrence Island, rather than Zaliv Lavrentiya, at the south entrance of Bering Strait. The 'Kamchatka Sea,' of course, would be what is now called the Bering Sea. What the poet meant geographically by Arctic Sea or Arctic ground is open to question- it could be the Bering Sea, the Strait area, or the southern part of the Chukchi.
That whalers never penetrated far until some time later is pretty well shown by the fact that Wrangel Island was not 'discovered' and named until the whaler Long found it in the late 1860's.

I agree that the song is probably post-1850 and after the bowhead became an established target; my objection is to the attempt to 'date' the song from the 'geography' in the text.
Lacking the original poem, how accurately the extant versions reflect the text of the author also is unknown.