There are number of basic and highly misleading confusions of the relation between moral subjectivity, moral objectivity and science. We can bring this into focus by noting that there is a crucial distinction between looking at "moral subjectivism versus moral objectivism" compared to other senses of "subjectivism versus objectivism". These are different senses and we need to identify these different senses to avoid a range of problems.
Epistemic Objectivity and Subjectivity
In empirical disciplines in general, it has been found that the most reliable and robust ways of understanding the world, the methods that best eliminate errors and minimise mistakes -the errors of reason and mistakes of fact - is best done by transcending one’s subjectivity - neutralising one's private practices, point of view, preferences, prejudices and perceptions - to aspire to and formulate a neutral, public explanation of the phenomena under investigation, one that is accessible to anyone regardless of their particular private perceptions and so on. This is a challenge of knowing - epistemology - so one can call this “epistemic objectivity”. By contrast someone who relies on or makes claims about features under investigation that are not, at least in theory, publicly shared in this sense, that is dependent upon some preference or perception that is private (and biased) or not generally accessible or available by others, this form of knowing can be called "epistemic subjectivity".
The results of what is discovered though epistemic objectivity, are called objective for short, as the results of what is claimed via epistemic subjective are called subjective. The later is also sometimes used as a pejorative term, indicating a lack of interest or avoidance of aspiring to the more reliable and robust, hence more justified, answers provided by epistemic objectivity.
However it is important to remember the sense of objective and subjective being used here so I will qualify this with epistemic prefix to avoid future confusion.
There is no reason to suppose in advance that investigating the problems of voluntary human interactions cannot also be be investigated scientifically, that is emphasizing epistemic objective methods over epistemically subjective ones to perform and derive objective conclusions in this domain.
Moral Objectivity and Subjectivity
However when looking at this domain - of ethics - from the inside rather than the outside, the terms subjective and objective have different meanings or senses which can cause great confusion. Within ethics, objective theories are those that are mind-independent and subjective theories are those that are mind-dependent - by definition.
Now let us combine these two senses explicitly to try to dis-ambiguate the confusions that can result
1.When performing an epistemologically objective analysis over the problem of voluntary human interactions, one cannot avoid investigating and using as part of the data set, the fact that there is, at least partly, some sort of mind-dependence being studied. That is the target of the field being studied, from the inside we can still grant that this called moral subjectivity. Still there is no reason in advance was to why this is not possible, that is when we combine the differing sense of these terms we can say that the challenge is to epistemically objectively study moral subjectivity. So any argument that endeavours to show that some form of mind dependence is involved such as reliance on beliefs and desires and proceeds to label this as morally subjective says nothing about preventing an empirical investigation of the data, that is an epistemically objective analysis.
2.We can go the other way on this. An epistemically objective analysis can be classified differently from within ethics. Since an epistemically objective analysis investigates, at least partly, brain states, a theory or model formulated based on this could be called morally subjective. However since it does not exclusively do this and emphasizes and includes the material and physical interactions between agents, it could alternatively be called morally objective. Supporters and critics of different position within ethics might label such a model produced by such means as suits their purposes. Such labelling makes no difference to the soundness, strength and validity of the model.
3.Within ethics, the distinction between moral objectivity and moral subjectivity can become a (arguably) false dichotomy if it is asserted that these as mutual exclusive and (not quite- unless one adds nihilism) collectively exhaustive. Whether, or not, moral subjectivity and moral objectivity are the only two games in town, this says nothing about the issues of epistemically objective investigative ethics.
4.The fact that within ethics one might, most popularly, only see two positions, the false dichotomy alluded to above, say one that includes mind states and excludes everything else, stereotypical subjectivism - morals as dependent only on belief and desires - versus moral objectivity that specifically excludes this such mind dependence as in intrinsic prescriptivity or natural law theories, is only a view on the internal definitional practices within this field. It is possible that an epistemic objective analysis can show that both are flawed and mistaken and that it can propose new models that do not fall simply within the popular classification within ethics. If, from within ethics, there is still resistance to seeing any such alternative this is a problem of the internal definitional practices and it is these that need to be reviewed, revised and possibly replaced or rejected.
5. The fact that human subjectivity is being studied does no deny the possibility of a successful epistemically objective anlaysis. Part of the data under study is that aspect of human subjectivity to do with, at a minimum belief and desires. These are quite ameanable to empirical investigation. Whether one uses (neo)Humean or Frege/Russell belief-desire psychology, Belief-Desire-Intention (BDI) theory, 1st person intentionality (Searle), 3rd person intentionality (Dennett), even eliminativism (Churchlands) or some other brain-mind theory this still applies. Such subjectivity is not a problem for science and so not a problem in investigating the field of ethics either.
A controversial last confusion
I think the above five points are straightforward, the following is the most controversial one and needs a full post devoted to it but I will assert it here for completeness.
What is being studied and what, if successfully would be developed, would be a science of prescription. The challenge is not only to describe how these prescriptions work, their origins and their limitations but also formulate hypothesis about prescriptions and see if they work in reality, to review, revise, replace and reject models to support and push for the best provisional and defeasible answers. There is no reason in advance to say this cannot be done not even in principle. There are certainly some common - obvious and subtle - rational and empirical traps that need to be avoided and often are not, but these are insufficient to prevent this enterprise at all. The only way to make that case is to show how this field is somehow uniquely special compared to all and any other field but then one has to use epistemically objective methods to make such a case, but that requires assuming it is not uniquely special and amenable to such approach as is any other field. That is I fail to see how the denial of the possibility of a science of prescription can be made without contradiction.