In addition to these sort of hypothetical mock-ups -- that we can test as the evidence is presented-- I have an idea how to move forward. I propose we trace the trajectory of chanties by presenting information from each decade. We could start with the 1810s (although there will be very little). Confining ourselves to that decade, we cite references to chantying generally, specific songs, etc., perhaps along with relevant details of geography, demographics, and technology of merchant sailing.

To start off the 1810s:

LANDSMAN HAY, though not published until 1953 (?), consists of the memoirs of one Robert Hay, 1789-1847. In them, Hay is supposed to have reported seeing/hearing stevedores in Jamaica in 1811. To me the reference (which I've not seen first-hand) is vague, but it seems like they are using a capstan to work cargo. Hay notes their in chanty-like song "Grog time of day," which turns out to have been a popular song associated with the Caribbean region through the early 19th century.   

Grog time of day, boys
Grog time of day,
CH: Huro, my jolly boys,
       Grog time of day

[I don't know if the chorus marking is in the original. I've taken this from Hugill. In other references to this song, this whole bit makes up the chorus]

It's possible that such a form was at that time distinct (or fairly distinct) to either the specific region or the specific ethnic group (Afro-Caribbeans). I say this because the way in which it is described, it is as if only "others" (with respect to the author) were engaged in the practice.

For some framing context, the packet ship trade began after the War of 1812 (I don't have specific dates), with the Blackball Line for example starting in 1816. The packet ships are thought (e.g. Hugill) to have necessitated a different way of handling work with smaller crews.