Territorial isn't a fair descriptor. That implies that they are hostile or at least quite wary of people entering their territory. But the We have been remarkably welcoming of people into their territory for pretty much the entire time we've known them, and have not responded with aggression or lashing out at any point. The closest we get to that is the rare occasion when a lone hunting We-spider is in its hunting grounds and a dwarf wanders too close without warning, and even that's because it's separated from its greater hive-mind and acting on instinct to being surprised in a place where it's expecting danger/hostiles.
The Library-We are specifically welcoming of visitors and trained to assist visitors, not be wary of them.
I will grant you that the giant spiders would be something the casual library-goer would avoid, but KAU boasts a pretty staggering selection. Anyone willing to make the trip to K8P would not be put off by the giant spider librarians that even the famously conservative dwarves have no issue with. And that "no issue" isn't from lack of exposure, either: dwarves from throughout K8P have contributed towards building the bookshelves, tracks/rails, carving out the library's spaces, transporting books into the library, affixing runic lights and furniture, collecting silk, and more.
For academics who want the chance to read books from the ancient Library of Mournings, books on seemingly any subject from throughout most of the Old World, or tomes on magic (normally not something you can find in public libraries, except for KAU), and an extensive selection of dwarven books, braving the friendly giant spiders is a challenge they are probably willing to tackle for that otherwise-inaccessible fruit.
And while they're there, they can study and get material for academic papers on the human population (and halfling population!) living in and being part of a full-blown Karak. Now that the Karak is fully secure, the river/road to and from the Karak is secure, and the canal linking travelers safely all the way from the Empire to Karak Eight Peaks, there's no danger to worry about, just the travel time and expenses.