There are plenty of traditional songs in the English language which involve transformation into all sorts of mammals, fish, birds and reptiles, but the question was specifically about lycanthropy. Seals can look a bit like dogs facially, but that's stretching the point rather too far!

You'll find "black dog" stories current to this day, but no old songs about them that I can think of. Wolves appear in plenty of songs, but they aren't people: they are very much other.