Wow, I forgot completely about the Hagia Sophia.
Maybe Layer 2 is simply one of those places that's a direct mix of two different cultures. (Like the 32 Chambers)
The building we see could easily be just a hybrid of a mosque and a basilica that has church-like structures on the outside and mosque-like structures on the inside. The arches and walls are clearly Islamic, but the dome and the floor are clearly those of a church.
EDIT: However, I do not withdraw my statement.

I still think "Cardinal Mosque" is appropriate and give equal representation to both cultures involved.