I'm only 10 years late to the discussion, but stumbled onto it last night when, well, researching Mattie Rag. I have some thoughts pieced together after listening/trying to decipher 10+ versions.

The earliest recording with a definite date that I found on YouTube is a version by Lord Flea with The Jamaican Calypsonians on the Times Record label from 1952.
https://www.youtube.com/watch?v=YEdGbZKfzkU

There is a cool version by Tony Johnson from an album called Calypso in Britain 1950-55 with additional verses about a glamour gal, Mattie, who wraps her hair in a rag, and who he takes on a plane to Paris. The standard verses are also contained in this version.
https://www.youtube.com/watch?v=V0Yif0pIrGk

Someone mentioned the similar song Charley's Cow. I found this mentioned in a Catalog of Copyright Entries: Library of Congress, registered in 1957 to Edward Seaga. By the timeline, Charley's Cow is clearly based on the earlier Mattie Rag.
https://books.google.com/books?id=FzQhAQAAIAAJ&pg=PA67&lpg=PA67&dq=charley+cow+song&source=bl&ots=L7P47JBphA&sig=ACfU3U22J05t5wJ1pSw0gWHqFLmKj8F7BQ&hl=en&sa=X&ved=2ahUKEwje9fCT073xAhXvlmoFHRMnB9gQ6AEwEnoECBoQAw#v=onepage&q=charley%20cow%20song&f=false

In some versions it's a "gal" who chew a rag. In some versions, a "dawta". In a couple, a dog. In the versions with a female(s) chewing the rag, it feels like wordplay with "chewing the fat" and a physical rag that belongs to Mattie, who I don't have a firm idea who Mattie is.

Regarding the versions with a dog a chew on rag, I found this Jamaican "proverb" - "Dawg have money nyam cheese, wen im bruk im chaw ole rag." at this site.
https://jamaicajamaicawi.wordpress.com/2021/02/10/jamaican-proverbs-from-a-z/

I think someone suggested papa might have gone down a mango walk for a carnal liaison, but it feels weird that the son who successfully ran away would have accompanied papa on such an excursion.

Someone mentioned the tradition of a mother/grandmother praying on a rag and giving it to a son to protect against capture into slavery. I found that very interesting. However, in both Charley's Cow and the much later Whatcha Gonna Do by Peter Tosh, there is a clear criminal offense mentioned for which papa is held.

There was talk about how the sing sweet songs and play guitar seems too light in spirit for the context of papa having been captured. It sure does. I have no speculation better than anything I read above to offer on that, except that perhaps in the context of the song Mattie Rag being sung in real time in real life it's simply an invitation for everyone to join in for the chorus.

Overall, my feeling is that there may not be one clear, correct meaning to the song. In some versions the version "dem catch papa" is first, in others the verse about "who fa rag" is first, making it seem like they aren't really connected, just two verses to sing, the order unimportant to any story being told.

Sometimes the mystery of the journey and the seeking is the greatest gift.