What is the correct translation of lucky bird into French?

The literal translation, like for the iphone game, is "oiseau chanceux", which does not mean anything really in French.

The translation of the expression "lucky bird" meaning a bird (and by extension, a person) with its belly full, does not exist.

There are however several adjectives or expressions to describe this physiological state, but I don't know if this is what you want. Here are some: repu, rassasié, avoir le ventre plein, or from a famous song, avoir la peau du ventre bien tendue.