I would use the Relevanssi plugin which allows you to search PDF files. Upload the PDFs to the media library. This will make them attachment “posts.” Relevanssi can now index them. Make sure that Relevanssi will display PDFs in the search results.
As for importing your Calibre library, you might be able to use the plugin you mentioned to create a blog post (or a custom post type) with the PDF files attached/included somehow. Otherwise, if it’s possible to export all of your e-books from Calibre as PDFs, you could do that and upload them all to the WordPress media library. If you want to be able to write a description for them, you’ll want to make sure that whatever theme you use can display file descriptions.
Edit: Here are two plugins that I found just by searching “document library” in the WordPress plugin repository. They both appear to be free, but they might not be what you’re looking for:
Also, if you’re planning on using WordPress only as a document library, you might want to consider using software designed for document managment. Here are two free ones I found:
You’ll have to host them yourself. Depending on what hosting provider and plan you have, you might be able to use the same hosting.