I have little interest in ethics (type, not token!). I only have one book by Singer, but his philosophy comes from a scientific preoccupation with the nature of consciousness in animals as a path to understanding morality as his advocacy for more legal rights for animals attest. Since you invoke Singer, I'll approach the question from a naturalized epistemology on this as he does the same. I'll sketch what I believe to be a "generic consensus" of grounding self-awareness as fundamental to the manifestation of sophisticated morality. I personally believe that consciousness isn't entailed in any strict sense by self-awareness-as-theory, so I'll let that matter to others.
Is there a good, firmly grounded justification for valuing self-awareness over other kinds of awareness, either as essential to consciousness or from a moral perspective?
Yes. Human eusociality reaches a pinnacle in mechanism in the form of morality which is a blend of animal religious behavior founded on psychological altruism combined with human reason and language use which allows for a gradation of moral expression culminating in the application of universal principles such as Kant's categorical imperative. In philosophical terms, one might consider there being a spectrum of morality from pre-linguistic emotivism to a fully cognized ethic. One such example of a gradation is Lawrence Kohlberg's stages of moral development.
Instead of rehashing Singer's positions downwards into the grounding theory, let's start with the notion of animal proto-religiosity and work our way up, tracing a semi-reductionist view of the interrelation of ontologies from biology, up to psychology, up to ethics. (This, of course, is no surprise given his thoughts in Consilience.)
First, we start with sociobiology in the form of Edward O. Wilson, the founder of the discipline, whose book On Human Nature, reformulates his earlier work for understanding how biological teleology (SEP) can be used to explain altruism (recognizing a distinction between biological and psychological varieties). His other fascinating book, The Social Conquest of Earth, shows the consequences of how altruism impacts social organization. This is a tremendous accomplishment because it serves as a bridge between morality and evolution, especially because E.O. Wilson came from a religious fundamentalist background and read the Bible cover-to-cover carrying his fascination with morality into the hard sciences. But, there's a better body of work to show more immediately how morality is grounded in self-awareness, and that is the work of a noted primatologist, Frans de Waal.
The most thorough expression I've found on this matter are books like Primates and Philosophers and The Bonobo and the Athiest by de Waal who is a preeminent primatologist and reflects a lot on the origins of morality. From The Bonobo and the Atheist on page 235:
Our moral codes apply fully only within the group, be it a language group, a nonliterate population that shares the same piece of real estate or the same ethnic identity, or a nation.
This quotation is an obvious recognition of in-group-out-group moral thinking, and extends from the logic raised by sociobiologists in regards to how biological altruism leads to psychological altruism. Both books, as well as others like his book on Chimpanzee political behavior, make for an extremely compelling argument that dominance hierarchies are more maintained by altruism than by force. The best chimp alphas, for example, are peacemakers and behave in such a way to unify their bands by behaving fairly. Unless one has some academic blindness to animal behavior, this is consistent with human behavior.
So, from biology to altruism is half the journey. We can trace a philosophical arc citing literature on intentionality and the philosophy of language, most contemporaneously through Searle and Tomasello, but I'll presume you have some familiarity with these philosophical notions broadly. What is important to take from their work is that with the appearance of consciousness and communication, language steps in to provide a mechanism for the exchange of information about mental states, which exists for all except the strictest of behaviorists, or course. (Yuk, yuk.)
This lands us squarely in ethics, which is seen by de Waal as the application of cognitive processes (in the narrow sense) to emotional impulses of altruism. Therefore, systems of ethics don't just explain intelligent behavior, but are arguments that justify philosophical action (being no small irony that epistemologists sometimes borrow ethical theories for forging analogs). These sets of ideas are ontological correlates, right? Those of mind and body. Now, the variety of ethical positions is a book in itself. What's important to think about is to what extent ethics and morality are grounded in psychology, and countless psychologists explicate the topic.
Zimbardo's Stanford prison experiment was an illuminating incident in moral flexibility, and even has a decent movie made about it, but perhaps one of the more famous psychologists who has explored morality specifically is Lawrence Kohlberg who had a taxonomy of morality starting right here in Chicago. From WP's article on the stages of development:
Level 3 (Post-Conventional)
5. Social contract orientation
6. Universal ethical principles
At stage 6, the pinnacle of morality, Kohlberg ties together ethics and conscience both of which are most sophisticated when articulated from a perspective of self-awareness, which might be most famously articulated in the form of the Golden Rule.
You asked for a justification for the importance of self-awareness to consciousness (I declined) and morality, and where we end is where morality and self-awareness intersect, from the WP:
The Golden Rule is the principle of treating others as one wants to be treated. It is a maxim that is found in most religions and cultures.1 It can be considered an ethic of reciprocity in some religions, although different religions treat it differently.
And it's tough to reciprocate if you cannot reflect, because as Dunning-Kruger has shown about consciousness, it's difficult to model a theory of mind of others, particularly moving past a folk psychology. Self-awareness, then, maybe the primary mechanism by which to build a pragmatic set of universalist principles of ethics, as Kohlberg recognizes. From there, whether you prefer Spinoza, Confucious, or the mighty Kant, it's merely a disputation of metaphysical presupposition.