Now, that wouldn't be too difficult, but if Squeenix were to outdo themselves as they usually do, I could see a /beckon providing a piggy-back/shoulder ride.
I could see them giving you missions or quests in the form of requesting food or a toy or a specific item. Successfully completing each task would fill a happiness meter,
Which Gives EXP when full.

Equipping glamour gear made for children, or using some of the available glamours, would also be a possible idea, mainly so that it's your child and they're not wearing the same thing as everyone else