The present Swabian Defence of Moral Realism is a complex (though I hope, not too complicated) structure of arguments and ideas. Which of them you may value as 'sticking points' will obviously depend on which debates and arguments you have as yet directed your attention, and which general meta-ethical, ontological, methodological, epistemological (etc.) views you tend to assume. Here is a list of those elements of the Swabian defence which seem to be most likely to be found interesting by some.
(1) The truth or falsity of Moral Realism need not be an a priori matter. If so, the search for a strict 'proof' of Moral Realism may be just an unwarranted, inadequate requirement. Thus, the existence of moral facts may be grounded in the experience of suffering and pleasure. Yet the existence of these psychic states may be (if a fact at all) a contingent fact, to be established empirically.
(2) For Moral Realism to prevail it is not necessary that a strict proof (can) be given. In general, being a strict proof is not a precondition of an argument's success. If an argument shows that the truth of Moral Realism is much more plausible, or much more probable than the truth of Moral Non-Realism, all other things being 'equal', to endorse Moral Non-Realism would be just a mistake.
(3) The denial of Moral Realism is a 'radical' view. (This, to be sure, is not meant as a pejorative assessment; yet we must be aware that ...) To deny Moral Realism you must deny the existence of any single moral entity. (And on the other hand, nota bene, to reject Moral Non-Realism you need not demonstrate the correctness of any particular moral system, or the occurrence of any peculiar moral entities.) The occurrence of any old single 'moral entity' falsifies Moral Non-Realism. Yet at the same time, some moral entities are rather 'easy to get'—consider only the definition of 'moral intentions' given below: the term refers to certain are common-or-garden intentions.
(4) Often, partial definitions are fully sufficient to enable an argument. Thus, for instance, to demonstrate that certain intentions are 'moral intentions', and certain actions 'moral actions', you need not provide full analyses of these terms; it is sufficient to establish adequate sufficient conditions for these terms to apply, and then to demonstrate that these conditions are satisfied in certain cases. Or, for another example, to defend effectively the existence of 'oughts' you need not have a complete analysis of the notion, or of the extension, of 'oughts'; it suffices to give sufficient conditions, and to demonstrate that these conditions obtain.
(5) 'Subjective' states may exist objectively, and be objectively (dis-)valuable. If a subject is in a certain state, and the subject itself exists objectively, then (given that states can exist objectively at all) the state will exist objectively, too. That a given state is the state of a subject does not imply that the state's existence is subjective (not objective, not real). There are two senses of 'subjective' which must not be confused: (a) being one property of an entity of the type 'subject'; (b) existing not really, but only on the views of a certain subject, or certain subjects.
The Swabian Defence suggests using the so-called 'hypothetical-world-part' tests to establish objective existence and objective (dis-)value of certain psychic states. It thereby presumes our large familiarity with certain psychic states (suffering, pain, ...), and it accordingly assumes that our opinions about them are evidentally significant. If the tests work the way they are meant to, they reveal that we represent those psychic states neither as mere illusions of subjects ('existing subjectively'), but as existing in the hardest sense, and not just subjectively (dis-)valuable, but indeed objectively (dis-)valuable, (dis-)valuable even from an impartial point of view.
(6) Contrary to widespread doubts about the existence of values, the existence of (dis-)valuable states is apparent: psychic states like suffering and pleasure are clear and apparent examples.
(7) That someone can create a valuable state or prevent a dis-valuable state is sufficient for the occurrence of a 'moral ought'.
(8) The existence of a moral intentions, 'moral oughts' and moral actions is sufficient for (amounts to) the existence of moral entities.
(9) As it thus turns out, it is unworldly to deny Moral Realism (though not by means of logical or semantical necessity).