what are we supposed to say again when christians ask us where we get our morality?

[Neo-Scholastic]

Why are you talking about empathy in isolation? Why are you ignoring the other genetic traits of reciprocation and a sense of fairness? Why are you equating this directly with breeding success?

Maybe because you do. As you also said...

Societies with deeply flawed rigid moral systems don't survive long term and are replaced. Societies with moral systems that allow development tend to do better.

The theory of biological evolution is about, and only about, reproductive advantage. In terms of evolution, empathy, reciprocation, and fairness have only survival value. For other values, like moral or aesthetic values, people must look elsewhere.

We are intelligent social creatures. We have evolved to strive to work together and this carries biological advantage. The tools we inherit to do this include empathy, reciprocation and a sense of fairness.

I agree. Human beings have all these features. Humans also have opposable thumbs, color vision, and walk upright.

From this we have developed societies with complex moral systems - some better, some worse.

By what basis do you determine which moral systems are better or worse? If you say that one is more fair or more just than another, then you are saying that one is morally superior to the other. Doing so either relies of self-reference or an appeal to some meta-morality system (which means the original systems being evaluated were not sufficient). The question remains unanswered. Humans may make rules of behavior but what is it about those rules that makes them moral? Having the capacity to make rules/codes is much different from those codes/rules actually being moral.

The OP asks if there are non-theistic foundations for morality. Looking to evolution fails.

...there are many behaviors which have survival advantage which have nothing whatsoever to do with morality ..

My point exactly. The common feature of all evolved features is that they confer a reproductive advantage. To have an adequate moral theory one needs a way to distinguish those evolved features related to morality from those that are not. And that requires that you have a moral standard already in place that you can use for making such a distinction.

We point to evolution as giving rise to moral feeling and choices.

There is some ambiguity in this statement, but I will try to address what I take it to mean...A wide range of impulses, instincts, dispositions, and abilities contribute to moral choices. A moral system serves as a guide for balancing competing impulses, like current desires versus long-term benefits or emotional responses versus rational reflection. Evolution gives you the underlying considerations, but not the ultimate goal of the moral system nor the best solution for achieving that goal.

Morality only has purchase within a society of like-minded individuals.

More precisely, the implementation of a behavior codes/rules regardless of whether is comes by decree, by social compact, or individual conscience is not grounds for determining whether one system is better or worse than its alternatives.

Personally, I am not fond of “moral systems” because such systems end up being either self-referential or devolve into basic compulsion by force. I see it more as a matter of what someone considers most to be desired and choosing wisely to achieve that desire.