Moral realism has no epistemic superiority. It's just as "bullshit" as everything else; the rotten core of bullshit is just hidden under a bunch of layers of deduction.
I prefer to air the dirty laundry; realists are uncomfortable by stained underwear and would prefer to play logical games on top of an edifice without thinking too much about the sand underneath their feet.
But at the core all systems of morality have the same problem. Realism is no escape. In fact, IMO, it's worse than a mistake because it provides the illusion of solving an insurmountable problems.
I guess it comes down to this: "I choose to believe in God and I know that belief is arbitrary" is to me by far the most compelling apology. Things like Goedel's proof just make me yawn and squint.
It's built into our everyday physical reality, or it is bullshit.