Origin of "Hark, hark, the dogs do bark, the beggars are coming to town"

This says it’s about the dissolution of the monasteries. Is this claim credible?

From here:

From here:

Unless Cecil is willing to dispense the Straight Dope on this one, I’m not sure if there’s a real answer to be had.

From that site:

Funny how Peter and Iona Opie, the authorities on Nursery Rhymes, can only date it from 1672. It could, of course, go back farther, and probably does. But at least give me a cite for the 13th C.

From Heavy Words, Lightly Thrown, by Chris Roberts (a book which dissects the history and origins of many nursery rhymes):