Is either the five-story pagoda or three-story pagoda allowed to climb up through the stairs?
No on going up.
They arent meant for viewing that way, mores the pity.
As far as I know, both the 3-storey pagoda and 5-storey pagoda co-exist at the temple area but you are now allowed to climb up the top. Only the National Museum and the Eastern Golden Hall are open to the public upon payment of entrance fee. Nice to take pictures from outside. Have a nice trip.
Get quick answers from Kofuku-ji Temple staff and past visitors.