It's a very intriguing question.
I think to answer it well and carry the analogy well, we need to distinguish between two uses of the word "language" and think about how they relate to the ideas at work in debates about moral realism. One a certain level, language as in "French language" refers to something that a group of people use to communicate. Clearly, the specific rules and conventions of this language are different from say the "Romanian language" or "Cantonese language."
On a certain level, there are clearly no objective (and eternal) facts about "language" in the terms of a single specific language. Grammar, pronunciation, and vocabulary all change over time. Maybe at any given moment, we could say French has some specific set of these rules and practices.
A second meaning of language which I think appears at some points in your question is the idea of language -- not as the specific linguistic practices of one group but as the features necessary for any language to work. I.e., are there "facts" about language in general.
Morality too can bleed between these two uses (some authors use "ethics" for the systems of particular groups and "morality" for the universal type -- others the opposite; the usage is entirely inconsistent between authors and translations).
I take it that the debate about moral realism is primarily about the second type of usage but bleeding into the first.
Looking primarily at the second sense of language, I take it that Chomsky and others are committed to linguistic realism insofar as they identify deep features of languages that they think are quintessential to its form. Others will also be linguistic realists if they believe that languages relate to communicating about and manipulating a natural world that has objective features. (i.e., any language is a language precisely insofar as it enables us to talk to each other about this world, so this world-fit will be a feature of language per se).
Linguistic anti-realists would tend to be post-structuralists and deconstructionists (e.g. Derrida) and others who think that the language blocks we're manipulating don't map onto "a real world" or "given" in any deep or meaningful way.
In the second sense, I don't see late Wittgenstein as necessarily being a lingustic anti-realist. To acknowledge language is a tool that works in certain contexts implies that language works. Whether he then is a lingustic anti-realist hinges on whether he thinks that it works only in an imaginary play-land or in the actual world which would require words to at times pick out and meaningful link to the world. I'm not an expert on reading his journals (most of the writings outside of TLP and PI), so I don't have a strong opinion either way, but I would tend to read him as a realist about language in the second sense of "language" which I take to be a better parallel to moral realism than the first sense, which I take would be akin to moral philosophers debating whether the moral system of Finland is the one true moral system.