This was a paper delivered by Dimitra Fimi at Tolkien 2019 (during which there was much singing) which I will attempt to condense briefly (though I highly suggest listening to it for yourself. I begins discussing Tolkien's reusing of the Poem. It was originally published at Leeds as 'The Root of the Boot' in 'Songs for Philologists'. This version contains more dialect words and religious references (Tolkien took much delight in singing of Tim "If Bonfire there is 'tis underneath"). However, Sam song is not only a direct recycling of the lyrics but also a much older folk tune. Dimitra goes on to show us the origin of the song in a tune known as the 'Fox went out'. In this song the fox manages to feed its offspring by stealing a goose from the farmyard. This makes us think of the fox sympathetically as we want to see its 10 cubs fed and get one over on the Famer. This was descended from the older Middle English folksong. However this is against the tradition of the sly and tricksy fox in much liturgical depiction instead being much more in line with foxes like Reynard. This talk also shows us other depictions of medieval foxes which I won't explain here. Now the Troll in Sam's poem is also a morally ambivalent creature (its munching on the bone wasn't doing any harm) as is the fox (it is sly and tricksy but we look at it sympathetically). This helps to unite the lyrics and the tune and Dimitra theorises that the Fox song is a real song in the Shire and that Sam has adapted it here. Now, Book 1 is often concerned with the folklore of the Hobbits and many concepts (adaptation of the tunes and lyrics such as Bilbo's walking song or the Hobbit's version of the Dwarves Song at the start of the Hobbit) of folklore and of the traditional foxes (Tolkien used the name of the Wolf Ysengrim the Wolf for the Isengrim Took). As such the fox "On Business of His Own' could be of to steal a goose for his cubs like in the ‘Fox went out’ and as such is steeped in the folkloric tradition of both Britain and The Shire. It is an ambivalent creature just as the Hobbits are crossing from the safety and good of the Shire to the dark and evil world outside. It "bridges the gap" between The Shire and our world while also allowing us to be reminded of the peace of the Shire chapters later when confronted with darkness and evil.
This is my brief abridgment of a paper which I highly Commend and implore all to watch or listen