Yes, that is quite close. And now that I have a better handle I can clarify: Eliminative materialism is not itself "false" -- it is just an interesting purist perspective that happens to be impracticable. The fallacy is when it is inconsistently applied.

Moral skeptics aren't objecting to the existence of morality because it is an abstract idea, they are objecting to it because the intersection of morality with our current logical/scientific understanding of morality reduces to something trivial compared to what we mean when we talk about morality. I think their argument is along the lines of if we can't scientifically extend morality to include what we do mean (for example, at least label in some rigorous way what it is we want to include), then we can't rationally mean anything more.