Either moral realism or moral non-realism is true so all we need to do is determine which position is more rational than the other. Moral realism is the view that moral statements can be factually true and moral non-realism is the view that they cannot be. If moral non-realism were true, then none of these statements could be factually true:
(1) moral progress possible.
(2) there can be a moral highground
(3) people's moral judgments can be incorrect
(4) in the case of two competing moral views on something, one person can be more right than the other.
(5) compassion is moral and cruelty is immoral
(5) fairness is moral and unfairness is immoral
(6) killing another person without sufficient justification is morally wrong
(7) punishing an innocent person is morally wrong
(8) raping an infant is morally wrong
(9) moral discussions are not a 0 sum exercise
(10) cowardice is of bad moral character and courage is of good moral character.
Based on aggregate trends in human behavior, throughout human history, none of these statements are indicated to be opinion-based truths. No justice system on earth, for instance, adheres to the principle punishing innocent people is morally good. Yet, moral non-realists would have you believe that this absurd conclusion, if it were accepted as true by someone, is just as rationally warranted as the opposite - that punishing innocent people is morally wrong.
So in short, the weight of the evidence (rationally, empirically, and prima facie intuition) strongly favors moral realism. In addition, moral realism is the prevailing view amongst academia and the public so in order to override the status quo, you must prevent a strong case to overcome that as well as all of the evidence in favor of realism. Until then, realism is the more rational position.