The photon here does not interact with the nucleus; it just needs the nucleus for momentum conservation. Why does the screening effect then effect pair production cross section? Also, why does the cross section increase with atomic number?
I think that the curvature of the field is larger for larger atomic number $Z$. Therefore it is easier in the vicinity of a larger nucleus. Then the screening effect takes away the field strength again.
Something has to interact with the nucleus, otherwise it would not be able to absorb the momentum. So a virtual photon has to couple to the new particles and the nucleus. If there are more particles in the nucleus to couple to, this should increase the cross section.