What is the best way to keep up with my club's physical library of books?

Don't! Many BetterBookClub companies have a physical library of old, new, and borrowed books, but there is rarely realistic justification for trying to keep up with books regularly borrowed, read, and returned. If you do maintain a common access physical library, just encourage responsible use and invoke a monthly or quarterly system to add an appropriate number of new and relevant reads on the shelf.