I feel like there's two or three different problems that you're asking about, all mixed up together.
First off, if your media relations folks have control of the account and are preventing everyone else from doing useful things, then you have a major administrative problem at your university, and need to solve that at an administrative level. As a business entity, the university should only need one business developer account (for publishing applications in the App Store publicly), one university account (for internal use by the students in software development classes) and one enterprise account (for internal use by the faculty and staff). If your university is big enough to be multiple separate businesses, or you want to argue with Apple that you are, that's an argument you need to have with Apple.
Second, why on Earth would your students want to publish applications through your university account when they could just as easily (their own developer account would be $100/year, and compared to tuition prices and/or equipment prices, that's nothing) do it themselves?
Third, "we have been told that one company can only have one developer account". By whom? There are different types of developer accounts, and it's entirely possible and feasible for one organization to have two different developer memberships for different purposes.