I'm not sure if this bug is restricted to only the Rat Quest, or all quests that require you to kill something for it to be completed.
Would probably also apply to Animal Companion, possibly summons.
But anyhow:
My Familiar killed the rat, and there was no completion update. Instead, I was forced to abandon the quest.
This has been a bug for even a long time in EFU. A player has to kill the NPC or the party won't get the journal update (to my knowledge).
Vlaid, that has been true on some quests, but not on others. And the quests where it was occurring, the DM's have usually tweaked it in some way so it didn't happen anymore. Not sure exactly what they've done to fix them, but perhaps such a tweak needs to happen on the Fat Rat quest? This is the 2nd time the problem has been reported for this quest, I believe.