I believe the most compelling theist response is that morality is "written in our hearts" by our creator, and that whether you believe or disbelieve in Him, you still act in accord with what is written in your hearts by Him. Animal morality provides a slight but easily surmountable objection, as He is responsible for what is written in their hearts as well. Now I may personally find the idea preposterous, but it at least has the virtue of being consistent with the data, and therefore would need to be ruled out on other grounds.
(A common theist analogy is that morality is like air. You can deny the existence of air, but you will still require its existence in order to deny its existence. It becomes something of an unfalsifiable hypothesis, which, I believe, would make the overall argument an informal fallacy, but then it's not clear what would falsify or differentiate the naturalistic theory from the supernatural one. [At least at first gander. I'm not an ethicist and so I stumble upon such questions more by accident than by design. I am a rank amateur in the field.])