Yes, pretty much any ethical system based on evolutionary psychology relies on the fact that "I don't care" is a lie, that our sentiments are not randomly occurring desires specific to the individual, but evolved motivators common to all humans (other than the psychologically damaged), whose purpose would be to promote survival in co-operative groups.
With the signals from modern society being so far removed from the society in which these motivators evolved, it is unsurprising that there will be be those who consider their rational self-interest to be best served by selfishness or moral ambivalence and these may well claim "I don't care", but unless they have found a way to undo a million years of evolution it is unlikely that they actually don't care but rather can't see how to successfully apply their desire to care without compromising other objectives.
Your description of sentiment as capricious and moral laws derived from it as hypothetical relies on an a priori belief which can be empirically dis-proven (to the extent that anything can be). Kant said of benevolence "... this need I cannot presuppose in every rational being" without any empirical evidence to back up his assertion whatsoever. This is the trouble with trying to apply ethics, or any practically applicable philosophy, written hundreds of years ago, there will almost inevitably be a priori statements which underpin the rational argument but which can be made more accurate with modern scientific knowledge. We've very much moved on from this kind of wild speculation being acceptable and that makes for much more useful arguments. If a modern researcher in ethics began their argument with "I reckon not everyone desires the happiness of others" without showing any effort to support such a statement empirically, their work would be rejected.
Ultimately, you have two choices, either you accept that our motives, though often contradictory, are broadly evolved and contained within our minds and so estimable by science, or you don't. If you do then "I don't care" is irrelevant because the person either will or will not care according to their biology whatever they claim their sentiment is and so any theory of ethics can proceed as if they did care, not on the basis of what they say, but on the basis of what the best available evidence suggests their sentiment actually is most likely to be. If you do not accept such a notion, then there is little point in comparing ethical theories at all. If right and wrong are not to be based on something empirical and the theory's success measured equally so, then a theory's utility depends entirely on your acceptance of it's a priori definitions. One could admire the elegance of a beautifully rational theory from there, or admonish the ugliness of a poorly constructed one, but neither would be of any use if one did not believe the a priori statements at their outset. From this point of view "I don't care" would be a sound rebuttal, but would apply only the the person making the statement, and would be of no use to anyone else hearing it.