Exeter University is open to the public, although they've just done massive amounts of work on it and I don't know how it's turned out.
It might be worth taking another look at your nearest Uni library, as I think quite a few are similar to Exeter/Durham, at least in the day and if you don't want to take out books.
My local university is Lincoln and I'm on campus every now and then as I have friends who work in their enterprise park. They have an excellent and reasonably new library but you need a student card to even get in since they put barcode scanner gates in when they built it. Their Web site notes this also :-(
That said, I might go and explicitly ask sometime, as there may be an unpublicised way around the problem.
It might be worth taking another look at your nearest Uni library, as I think quite a few are similar to Exeter/Durham, at least in the day and if you don't want to take out books.