These two are neither songs nor about tooth rats but a pair of Oz aphorisms meant as terms of disparagement;
"He was a flash as a rat with a gold tooth!"
and
"Cunning as a dunny rat!"

And your tooth fairy gave out quarters? The going rate for Oz tooth fairies was sixpence when I was a lad, though I gather inflation has, since, got to the tooth fairies too.

The notion of a tooth rat does ring a bell, very dimly, in the distant recesses of what passes for my memory though. I'll try and think harder.

Cheers, Rowan