Joe, this is completely off the top of my head but was not another name for a dandy or gaudily-dressed individual of Shakespeare's time a poppin-jay? Which surely must accord with 'papingo-aye' so parrot or peacock would fit the bill.

I'm more convinced by the French connection. Why would a cage become a tree? A pear-shaped cage yes, perhaps even with something resembling a tree branch inside it. Could it be possible that it became known as a pear tree because of the song?