If you complete the quest for returning the tiny girl's puppy using a wish for a cute dog, you will only get the FLG accomplishment message "brought back joy into the life of a tiny girl" if you haven't been to PC6 and discovered the cute dog corpse.

To test this, in exploration mode, you can walk around the wilderness beyond the 6 day deadline, wish for a cute dog and bring it to Terinyo. The quest will be completed, the tiny girl will give you candy and the FLG will properly show the accomplishment message.

If instead, you wait the 6 days then go to PC6 and find the cute dog corpse then go to the wilderness and wish for a cute dog, the quest will be completed and the tiny girl will give you candy but you do not get the "brought back joy" accomplishment message.

The accomplishment message in the FLG should be awarded for both instances.