What about the alligator purse being an attribute of wealth: expensive, fashionable, and presumably well-filled with cash? The child would be malnourished rather than ill, so that the philantropic rich lady is the only one who can help. "Pizza" has probably been patched in later, but may nevertheless convey the original meaning. Now you may ask why in some versions she is being paid instead of paying - that must be corruption (not of social welfare workers, but of folk verses).

Just a completely uneducated theory.