Plus however widespread it becomes in usage, it still conjures up in most people's mind its inescapable source - the property Star Trek. While it may have a bit of a life of its own outside the canonical copyrighted works, it is not separate from them and arguably is unlikely to ever be viewed separately from them. You don't even have the wiggle room you have with Elvish (i.e. Elf is a race that appears in many works); 'Klingons' come from a single source, and always will. While the language remains so tightly tied to CBS/Paramount's copyrighted properties, I would see it as theirs to control.
What I wonder is whether a constructed human language, even if originating for custom purpose, can be appropriated by enough people to become a legitimate cultural artifact of humans. Esperanto was constructed for general use, and no one was trying to prevent that, so the origins are different. But if enough people take up Klingon at varying levels of proficiency for general use, would it cross into this domain? Practically it won't; short of an invasion I doubt you will ever see hundreds of thousands of casual Klingon reading proficiency users and many many thousands of translations, like you do with Esperanto. But they might have an argument wrt/ the core speakers group eventually that it is comparable in size to core proficiency users of a language like Esperanto. Just not yet.