I think you need to take more steps back. What are laws? Can there be absolute objectivity, or what limits are there on approaching this?
Nancy Cartwright in 'How The Laws Of Physics Lie' demonstrates that truths are about abstractions, and can only be as true as the given abstraction or system of them, are valid.
I would contend objectivity is unreachable, it is only ever reified intersubjectivity. This is illustrated in the ancient metaphor of Indra's Net.
Happiness cannot be externalised and quantified as the utilitarians wished, rather it is a value judgement applied to causes of wellbeing, to things which support 'flourishing' or eudaimonia, the achieving of our potential, as previously established and reaching beyond it to define new potential (it is in the nature of minds, and evolution, to go beyond what is recursively enumerable inside a current system). Happiness is like a compass direction, or gradient, in navigation - it can provide instant information, but it is not a map, and it cannot give us the full picture about wellbeing. 'Ought' is map, 'Is' is local evaluation. We need history, and cosmology, map, to navigate.
I would argue morality, what is moral, is the product of both historical insights into what supports wellbeing, and aspirations based on ideas about future wellbeing. We require both a history, framed in terms of moral lessons, and a cosmology, based on our understanding of where our decisions are situated in their possible impacts or significance for the future. Historically, that has been called religion. Cultural myopia around the local dominance of Abrahamic ideas, has limited the understanding in modern (& post-modern) Western philosophy of the significance of different practices in history, and future scope of this sphere.
Durkheim who founded sociology as an academic discipline, had to contend with increasing understandings of the sophistication and literature of Vedic Hinduism, Confucianism, and Daoism, which profoundly challenged Western ideas about what religion is, and thinking about Shintoism, and shamanism in complex societies, no longer frameable like in Fraser's Golden Bough as an adolescent phase of society. Durkheim's definition:
"A religion is a unified system of beliefs and practices relative to sacred things, that is to say, things set apart and forbidden–beliefs and practices which unite in one single moral community called a Church, all those who adhere to them" - The Elementary Forms of the Religious Life: A Study in Religious Sociology, 1915.
at first appears uncontroversial. But, ideas like habeus corpus, or freedom of speech, fit this criteria of forbidding certain behaviours to the community, with the community bound together by heroes of these values, by statements of them and oaths to those, and by placing them in prominance as part of a given community's identity. And, making challenge to them or ending adherance to them, risk dissolution of the community. This is a kind of moral generalisation of the social contract.
Nietzsche understood that the challenge to the religious tradition of Christian Europe is of a different type than previously encountered, because it goes beyond a reforming of ideas, a shift in history & cosmology that encompasses and reframes the old ideas in a new system that has some additional scope. Instead, it has to be both a return to something older, hero narratives and exemplary individuals, and a kind of unravelling of their scope to create new capacities, that there is a great risk to social cohesion from the loss of a shared history and cosmology, and that dismissing or unpicking can result in nihilism for some, for increasingly many.. And their alienation from the social system. For the risk of decoherence and destruction of societies and specific capacities they have developed.
As well as issues of social cohesion, of communities being defined by adherence to a shared interpretation of history and social cosmology, I would also point to game theory, for understanding morality as objectively as we can. These approaches are not at odds. 'Sacred' values are higher level concepts, which incorporate histories of social learning. Frequently this has been about how psychopathy and narcissism in leaders, can create a free-rider problem, destabilising a society, ie game theoretical dynamics. So by organising a 'social immune system' against these behaviours, the complexity of the social system is able to continue increasing.
Joseph Tainter in 'The Collapse Of Complex Civilisations' describes how both hard military, and soft cultural power, allow civilisations that can solve new problems better to keep taking, and generating, more resources to solve future problems better than their neighbours/competitors, in the 'imperial cycle'. Until, there is a mismatch between disruption or perturbation, and capacity of a society to reform, typically into a state with higher costs or more fragility which must be balanced by more resources or resilience, or else over-reach - this has happened to every complex society of the past. We wish to believe in our era progress is inevitable. In Dark Age Europe, and much of Indian history, the opposite view held. Tainter argues civilisational collapse is not simply bad luck, but inevitable, and will happen to our civilisation or it's inheritors also.
It seems obvious that has a parallel to moral progress, and that the social reality that for instance makes slavery impossible, and the death penalty almost impossible, should not be considered permanent, or irreversible. Moral verities about them rely on social realities, which can change.
I would challenge the concept of any truly 'individual mind', by reference to the Private Language argument. But also point to a universality of awareness itself, the basis for intersubjectivity, as in our ability to go beyond Turing Machines into evaluating truth values beyond what can be recursively enumerated in rules - that is, through being 'strange loops', beginning wherever we are but seeking versatile resilient rules. They may sometimes converge, like in convergent evolution. But more often not, they will form a landscape of contender mutants.
Languages and their culturally shared concepts, implicitly embody salience landscapes, sifting out and highlighting what kinds of thing are pertinent, for a context, in a culture. Selecting for local & system 'fitness', or tendency to get replicated or be resilient. Like an evolutionary pathway, that compounds a set of abstractions and behaviours with an environment, into an ecosystem, a dynamic collaboration of these.
A new mutation or technology or social behaviour, might provide scope for powerful new capacities. Like say the development of mitochondria: that doesn't mean it is the only future way for all cells to be. Some prokaryotic organisms have developed a more efficient form of photosynthesis since plants split from the lineage. And our own blood cells don't have mitochondria. Similarly a moral system is not about what is 'more true' or 'more objective', but how it increases capacities of the social system.
Consider how the system of concepts 'game theory' allows an analysis of both ants, human societies, bacteria, viruses. A candidate system of historical explanation and cosmology, that explains more and allows more effective future action, we call: a 'better' theory. Not a final truth, or fully objective. Science has abandoned those absolutes. There is only, better. A gradient, towards increased capacity of a system of explaining the past & predicting the future - with cultural goals and salience landscapes embodied, but, an aspiration to get more universal, explain different cultures, religious practices, species, and look toward alien contact that may well shake all our ideas.
In so far as we share a language, and develop it together, we contribute to and draw from a collective intelligence. A moral community, with it's maintenance of it's cohesion, and capacities, is also a kind of collective intelligence. We can aspire to maximum generalisation and versatility of these cultures, these systems of abstractions. But there will always be other evolutionary routes - we may turn out to be dinosaurs, with the future inheritors of the world beneath our feet, or trilobytes dominant for a time but destined to leave no impact at all except biomass and gas exchange.
Morality is a system of heuristics to create higher-level abstractions about game theory, and buy-in to these have allowed social systems to increase complexity like new cell types, or multi-cellular collaborations. But there cannot be a best way to be, only better. And that lesson converges with our deepest insights: stay humble, respect other systems that support wellbeing, beware hubris.
And choose systems which support continuous improvement, rather than seem to be fossilising - because however successful a morality is at one time, it might leave nothing behind but a few strangely shaped rocks, if it cannot face the unknowable challenges of the future.. Get busy growing, or get busy fossilising, you might say.