The only way you can resolve this is if a concrete reference to the song turns up that predates 1837.

I would suggest references to 'Tally-I-O' wouldn't be much use as we already know this was in use prior to that date. It would have to be specifically 'Tally I O, you know'

I presume you are aware of the reference to the song in Thompson, p218. He got the song from 'an old songster in Buffalo, The American Vocalist'. He talks of wanting to roar it out as a chantey with a captain friend of his who likes chanteys. If you can find a copy of the songster, it very likely will have a date on it.