> Having been published earlier in the century by Ramsay and then anthologised to death …

Yes, by the time the parody was published in 1775, Ramsay's version would have appeared in several more editions of the TTM, plus Herd and Percy and maybe others. But in 1752, when we are told the parody was written, would it have been in anything except the 1740 and 1750 editions of TTM, (plus perhaps Irish and American editions)? Not to mention that BA was only one among a good few hundred songs in TTM, and perhaps not the most obviously attractive to readers looking for the new songs that Ramsay seems to make the main selling point of the book.

On the other hand, I don't think the parody is necessarily to be taken as evidence that the Ramsay version was widely known in Scotland in 1752. It looks to me like the kind of thing that wouldn't have been particularly meaningful beyond the narrow circle of those directly affected by the event it describes, so for it to be effective as a parody when first written it could have been enough for BA to be well known among the author's fellow officers.