It seems that Wrap It Up was written to be sung by Gino Lupari, although the writers were other people. The Belfast Telegraph identifies lyric author Neil Johnston as a Belfast Telegraph journalist, actually, rather than a member of the band; he is deceased now. Arty McGlynn had a lot to do with the production of the album "Barking Mad" which is something like twenty-five years ago now, but he was not a band member either, although he set Neil Johnston's lyric to music.

And then there is Mick Daly.
The Belfast Telegraph is one of several sources which report that the "Dog," in Four Men & a Dog, refers to musician Mick Daly; the Telegraph goes on to say that in his native Cork, Daly was/is affectionately known as "the black dog," and that is how the band arrived at the name. I am not as certain, as I ought to be, that Mick Daly is one of the band members on the album "Barking Mad," but I presume he is, that being the band's debut recording, right?
Anyhow, Daly was not long with Four Men & a Dog, ironically, he was one of the first to disappear from the lineup. Since then, Four Men & a Dog have had reunion concerts and appearances, however I don't know if Mick Daly was present for those.