I doubt I would require a permit for a play house. You could use:
102.3 Exemptions. The following are exempt from this code:
7. Nonmechanized playground or recreational equipment
such as swing sets, sliding boards, climbing
bars, jungle gyms, skateboard ramps, and similar
equipment where no admission fee is charged for its
use or for admittance to areas where the equipment is
located.
I would surmise the intent of the code is to not require a permit for this type of structure. However, if the owner requested a permit, I would issue one.