I don't remember any FF game having a rat killing side quest even at the beginning of the game, let alone at the end after you've killed gods. Shit... I don't even recall FF having side quests at all until, like, 7 with the optional boss battles for powerful materia.
Trails in the Sky definitely comes to mind. But if you browse Square Enix's catalogue, there's probably plenty that fulfill it. Final Fantasy stands out because it's a little less tropey.