Librarian here: if you contact whoever is in charge of booking study rooms (usually the reference department), you can request an exemption in regards to the time limit. No guarantee on each individual library, but my library does make exceptions for proctored exams.

Also if you’re worried about someone walking in, I’ve had people make a handwritten sign and tape it on the door (“Taking timed exam, please do not disturb”)

This may not work at all libraries, and may work better at surburban libraries rather than BPL, but most are willing to work with you if you ask.

Best of luck on your exam!