Lifesaver, I feel you. If the GA truly understands about NPC or NIC or even NPHC for that matter then there shouldn’t be a problem. The organizations ARE separated by sex and they OPERATE DIFFERENTLY. Of course, you can have an All-Greek Council too. They usually just coordinate general Greek activities like a big philanthropy or Greek Week. I know that some posters on here have these kinds of systems and can shed more light..
Anyway, the GB says that a College Panhellenic Association is formed when two or more NPC groups have chapters on a campus. Here is an NPC link of CPH area advisors. I invite you to get together with the sororities’ advisors and contact the lady for your state. The NPC Area Advisor can surely help the GA to convince the administration as to why this is needed.
http://www.npcwomen.org/college/c_advisors.php
Good luck!