Agree - you'd still need evidence to convince the powers that be.  To get
some data to that end, maybe some snapshot usability observation where you
sit down a handful of users at a computer (better yet, on a tablet/mobile
device) and ask them to find the library's website (ensuring that the
browser doesn't default to university.edu).  Observe their strategy - do
they go to university.edu and click around, or do they Google?  Or do they
use the university.edu search box (which is often a Google site search,
playing by the same rules)?

If your users are Googling, having that data in hand might provide an
argument that your page should be Google-able and you should have some more
visibility/control.  Or you could set up some dummy sites and Google bomb
yourself :).  (Joking.  Kind of.)

-Lauren