The texts of all 4 of those songs are actually versions of a different song that goes back to the 17th century, Master Title, Robin-a-Thrush (Roud 2792). I think what has happened is that the Wee Cooper verses from The Wife Wrapped (Roud 117, Child 277) have become attached to the Robin-a-Thrush chorus in Scotland some time in the dim and distant past. The result has been that we come across other quite different hybrids of the 2 songs. The texts are easily separated if not the choruses.
The sluttish behaviour of the wife in both is quite different (I can itemise these if required but they're easily spotted) and The Robin-a-Thrush song has no wife beating of any kind.
Bronson wasn't aware of this so you find a mixture of the 2 in his volumes.