I had always assumed that the 'clouting up' was using a cloth to polish the shoes (using the old plural shoen) and the inference was of someone crouching down submissively at the feet of another person, and that whoever the two were the happening was as unlikely as all the other pairings in the song.

One of the rustic players in a Midsummer Night's Dream represents the moon, with a lantern (lant-horn?) and a dog - if I remember aright. That, however doesn't give an antagonistic pair the wrong way round/never going to happen meaning, which Henry and the pope would do.