In the United States, a language, generally, is described as a "specification"; that is, as a set of facts. For a language, these facts would be a series of statements along the lines of "X is a word having such-and-such a definition". Facts, in and of themselves, cannot be copyrighted (or patented, for that matter). What is protectable is a specific "implementation". Works created to describe the language are themselves copyrightable, such as dictionaries and programming guides, as are works created in the language, like stories and reference material written for use by those literate in the language.
That definition holds, without much question, for "natural languages", developed organically over time by an ethnic or regional group as a part of their identity. Nobody attempting to argue that English itself, or even a modern dialect like Ebonics, is protectable by copyright will be heard in U.S. courts.
However, Klingon and Sindarian are not natural languages. They're invented languages, developed artificially over a short time by a relatively small number of people (as few as one). The argument can therefore be made that these languages aren't subject to the "specification test", because the facts they assert are fictional and exist to serve a copyrighted work of fiction. Just as, for instance, someone cannot publish an unlicensed Star Trek novel for a profit, using characters and worlds from Paramount's protected fictional universe, use of the language without permission for personal gain is copyright infringement.
The closest legal precedent to date is in software copyright law, Oracle v. Google, a case concerning a re-implementation (with some literal copying) of the Java API, owned by Oracle who bought the original creator Sun Microsystems, to create the Dalvik runtime used in Google's Android operating system. The trial court found in favor of Google in 2012 on the majority of the charges, including the one on the API itself, deciding that the API was a declarative specification (similar to a natural language) and thus not subject to copyright.
However, the Federal Appeals Circuit court reversed, holding under an older Third Circuit opinion, Whelan v. Jaslow, that the "structure, sequence and organization" of a piece of software, including an API such as Java's, was in fact copyrightable intellectual property and thus by copying the SSO of the Java API, even if it rewrote every line of code implementing the funcaionality behind it, Google was using copyrighted material belonging to Oracle. It remains an open question of whether a "re-implementation" of a given specification, as Google did by re-implementing the Java API to create Dalvik and the ADK, constitutes "fair use"; the Federal Circuit in Oracle remanded the case back to the District Court to hear those arguments and a decision has not been made yet.
If Google is found in violation of copyright for the use of the Java API itself, that means the structural elements of a software language, essentially the set of facts that various words exist in the language and have a particular meaning, are protectable. That could be used to draw parallels to any non-natural language, including Klingon, Sindarian, Dothraki and other fictional ethnic/racial languages. The same parallels can be drawn if Google is found to be protected by fair use, which is predicated on meeting a four-factor test:
- The use of copyrighted work favors the advancement of society over personal gain.
- The use primarily copies the facts and ideas of protected work, not their "fixation" or specific implementation.
- The amount of the copyrighted property used is a relatively small amount of the full protected property.
- The effect on the copyright holder's ability to expolit the value of the protected work is minimally impacted.
The use of an invented language like Klingon within "fandoms" that identify with the original copyrighted work could easily be judged fair use using this test:
- the usage usually isn't for personal gain but instead to "advance" the fandom by more closely matching the characters of the universe by speaking the language;
- It's the language itself (the idea), not anything the writers have written in that language, that's used;
- The amount of the language being used in any given situation is small (though to be fluent, you'd need knowledge of the majority of the language); and
- The owner of the original property typically isn't economically damaged by the use, such as having sales of his own material using the language reduced.
However, use of the language in a situation like the OP's, where the universe probably isn't identified or identifiable as being the original universe in which the languages were introduced, might tip the scales, because it is primarily for personal gain over furthering the existing fandom. Multi-factor "balancing tests" like fair use can be fairly broadly interpreted, and it's possible, even likely, a judge might rule against the OP even if the other three tenets are met if the OP's intention, as he implies, is to lazily rip off a language and paste it in his universe to avoid having to invent his own.
For the OP to be guaranteed to prevail in a lawsuit, the precedent must be established that there is no meaningful delineation between an organically developed "natural language" and a systematically defined "invented language".
Of course, the OP could very likely claim de minimis non curat lex. Literally, "of the smallest, no care from the law", a doctrine applied in many ways including discretionally by the judge to spare the court having to hear lawsuits over trivial infractions with little potential harm done.