I know what I am
And, what I am is a man
And...so is Lola!
Actually is, "but I know what I am and I'm glad I'm a man and so is Lola." That keeps the ambiguity going. Is Lola a man or is Lola glad that he's a man?
Your version would make it clear that Lola is a man, which would take most of the fun out of the song. :-)