Then, as I add each piece (cover, copyright, TOC, chapter, etc.) it also asks me to choose the Patron level.
PW allows you to lock any post from post type.This means that if a plugin or a theme treats some attributes for something as post types, then those also will have the gating option added by PW.
Ie, apparently the plugin you use treats various properties of ‘book’ post type as different post types. Cover, copyright, TOC etc seem to be recorded as post types. This would end up all of them being offered the option to be locked.
You shouldnt need to lock all of those. Just locking the final ‘book’ posts may accomplish what you need. But if any particular element of that book also needs gating like ‘chapter’ etc, you may lock them separately as well.
The page you linked gives a 404 error. Additionally, it seems to be using non pretty permalinks.
Patreon WordPress requires pretty permalinks to be turned on.