I assume they don't allow you to take the listening copies out of the building. I haven't looked into it lately but vaguely remember reading something like that a couple of years ago.
Naturally I'd never encourage anyone to break the law but if some scallywag with a laptop were to go there and rip the CDs, then upload them, I'd be absolutely outraged and would write a stern letter to the Daily Express. Ahem.
Birmingham: Archives, Heritage and Photography Service Library of Birmingham Centenary Square Broad Street Birmingham B1 2ND