I was trying to make use of Greene's phrase: 'inner chimp.'

I don't think it is Greene's phrase. I spent some time searching, and can find only one place where he used it - a 2007 RadioLab interview with Krulwich. I would be willing to bet that he was primed to use that phrase by the journalist. He doesn't even use the word chimp in the cited paper.

In any case, Greene's arguments are incoherent even by the usual lax standards of evolutionary psychology and consequentialist naturalistic ethics. He suggests that a consequentialist foundation for ethics is superior to a deontological foundation because 'consequentialist moral intuitions' flow from a more recently evolved portion of the brain.

Now it should be obvious that one cannot jump from 'more recently evolved' to 'superior as a moral basis'. You can't even get from 'more recently evolved' to 'more characteristically human'. Maybe you can get to 'more idiosyncratically human'.

But even that only helps if you are comparing moral judgements on which deontologists and consequentialists differ. But Greene does not do that. Instead of comparing two different judgements about the same situation, he discusses two different situations, in both of which pretty-much everyone's moral intuitions agree. He calls the intuitions that everyone has in one situation 'consequentialist' and the intuitions in the other situation 'deontological'!

Now, most people would object that deontology has nothing to do with intuition. Greene has an answer:

In sum, if it seems that I have simply misunderstood what Kant and deontology are all about, it’s because I am advancing an alternative hypothesis to the standard Kantian/deontological understanding of what Kant and deontology are all about. I am putting forth an empirical hypothesis about the hidden psychological essence of deontology, and it cannot be dismissed a priori for the same reason that tropical islanders cannot know a priori whether ice is a form of water.

And so, having completely restructured the playing field, he reaches the following conclusions:

The argument presented above makes trouble for people in search of rationalist theories that can explain and justify their emotionally-driven deontological moral intuitions. But rationalist deontologists may not be the only ones who should think twice.

The arguments presented above cast doubt on the moral intuitions in question regardless of whether one wishes to justify them in abstract theoretical terms. This is, once again, because these intuitions appear to have been shaped by morally irrelevant factors having to do with the constraints and circumstances of our evolutionary history. This is a problem for anyone who is inclined to stand by these intuitions, and that “anyone” includes nearly everyone.

I’ve referred to these intuitions and the judgments they underwrite as “deontological,” but perhaps it would be more accurate to call them non- consequentialist. After all, you don’t have to be a card-carrying deontologist to think that it’s okay to eat in restaurants when people in the world are starving, that it’s inherently good that criminals suffer for their crimes, and that it would be wrong to push the guy off the footbridge. These judgments are perfectly commonsensical, and it seems that the only people who are inclined to question them are card-carrying consequentialists.

Does that mean that all non-consequentialists need to rethink at least some of their moral commitments? I humbly suggest that the answer is yes.

Let me get this straight. The portions of our brains that generate what Green dubs 'deontological intuitions' are evolutionarily ancient, present in all animals. So Greene dismisses those intuitions as "morally irrelevant" since they ultimately arise from "factors having to do with the constraints and circumstances of our evolutionary history". But our 'consequentialist intuitions' are morally relevant because they come from the neo-cortex; a region of the brain that exists only in mammals and which is particularly enlarged in humans. Yet, somehow, he doesn't think that these intuitions are tainted by the contingent nature of their evolutionary history.