It is easier to build than to tear down. So, with that in mind, I want to focus on building rather than tearing down.
I have said that an answer to the theist's question of the possibility of morality without God has to focus on ways to reduce the suffering that it is perfectly within one person's ability to cause others in spite of any evolved altruism.
I suggested that this task is done by using social forces such as praise, condemnation, reward, and punishment to promote (malleable) desires that tend to fulfill other desires, and inhibiting (malleable) desires that tend to thwart other desires.
Do I understand that everything hinges on malleable desires? Does this leave room for instinctual action such as fight or flight?
Well, fight or flight are, to some degree, grounded on malleable desires. Environmental forces can influence not only the likelihood of whether a person fights or flees.
However, to get at your general question, there is certainly room for fixed desires. Fixed desires simply have nothing to do with morality. It makes no sense to condemn a person because he has a gene that causes him to act violently towards others.
Such a person is sick, not evil. We still have reason to prevent him from acting on his desires. What we lack is reason to condemn him for those actions.
How do you actually define morality itself? Must it be absolute and universal to be true morality?
Regular readers know that I detest the trap of talking about definitions as if talking about the thing defined. How anybody defines morality turns out to be irrelevant. If all moral terms were tossed out as garbage, I could still say everything I need to say about desire utilitarianism without it.
On this account, there are no absolute prescriptions. Morality has to do with relationships between states of affairs and desires, and those relationships shift over time.
However, the absence of absolute prescriptions is not a threat against objectivity. Many scientific facts change over time. One of the best examples of this is age. My age changes at a rate of approximately 1 year per year. I am not the same age today as I was a year ago.
However, there is still an objective, scientifically acceptable answer to the question, "How old are you?" The fact that relationships change does not mean that statements about those relationships do not belong in the realm of science.
As for universality, there is good reason to ask whether there are some desires that people generally have reason to make universal - and whether there are desires that people generally have reason to make universally extinct.
Moral terms tend to be concerned with desires that people generally have reason to promote or inhibit throughout the whole population. Eliminate moral terms, and desires that people generally have reason to make universal or make universally extinct will still exist.
There seems to be a presumption of sorts that simply condemning someone is sufficient to actually change a malleable desire. However, the field of psychology suggests that reliably changing desires is much more complex than delivering a simple condemnation. For example, people often redouble their positions when simply condemned.
There is no such presumption.
One point I would like to be here is that a person does not need to be condemned to have his desires molded through condemnation. He could be the witness of another person being condemned.
The other person need not even be real. The condemned individual might be a character in a story or parable. However, the condemnation directed at this fictitious character can alter the desires of those who read or hear the story.
These relationships of cause and effect are, indeed, complex. The better we are at understanding them the better we will be at using them effectively. Yet, however complex they happen to be, we do manage to promote and inhibit desires in others through praise, condemnation, reward, and punishment.