Is Kant treating the sublime consistently in the third critique and Anthropology? In other words, what are the differences being addressed in these empirical and transcendental inquires regarding the sublime?
Goodreau's book The Role of the Sublime in Kant's Moral Metaphysics referenced here should be a comprehensive analysis for your question. First regarding sublime in the Anthropology:
Kant's Anthropology is relevant to the thesis of the present work because it contains a discussion of taste from the standpoint of empirical psychology, i.e., from the standpoint of the human being as a phenomenon in the empirical world. Hence it will be useful to compare Kant's treatment of taste with the formal treatment given in the third Critique... As such, it is aimed at a much broader audience than the third Critique... discrimination with taste is distinguished from discrimination through mere sense perception because the former, which is the discrimination of the beautiful, is held to be universally valid while the latter is only subjectively pleasing... here Kant is looking at taste anthropologically rather than critically, which leads him to remark that taste applies only to the form of judging aesthetically and not to the generation of the objects in which the form is perceived... In other words, if the producer of art were the arbiter of taste, any and all productions of genius would be beautiful, and, contrarily, genius would be limited to what taste has so far conceived to be "tasteful."
While both the beautiful and the sublime are aesthetic judgments, only the beautiful belongs to taste. The idea of the sublime is not contrary to taste, however; Kant tells us that the idea of the sublime can and should be beautiful in itself. Even the portrayal of the evil or ugly can and must be beautiful whenever an object is to be aesthetically imagined, for example in a dramatic production, lest the portrayal arouses distaste or disgust... A feeling of astonishment arises when one realizes that one's comprehension cannot measure up to the grandeur of the sublime phenomenon. Kant describes this feeling of astonishment as a pleasant feeling resulting from the continual overcoming of pain... This effort to comprehend or measure up to the sublime Kant describes as an effort to elevate oneself... The sublime is a feeling of emotion and not an object of taste, but the artistic presentation of the sublime should be beautiful and not contrary to taste.
So clearly in Anthropology Kant analyzed sublime from social and psychological POVs about a central concept taste which concerns the communication of one's subjective feeling of pleasure or displeasure to another (the social element to taste). The satisfaction that results from the tasteful experience must be considered as generally valid not only for the experiencing subject, but also for anybody else, hence the ideal taste has a tendency toward the external advancement of morality.
As regarding sublime in kant's third Critique:
From the beginning of his career until the end of his life, Kant was concerned with the problem of morality generally and with the problem of moral motivation in particular. His earliest works link the sublime to morality; in the Observations as in the third Critique, the experience of the sublime promotes the consciousness of one's moral worth. Consciousness of one's moral worth is also conscious-ness of one's dignity, a dignity that is the result of our ability to transcend the spatiotemporal empirical world. This ability is our supersensible faculty, and it is this that we feel when we experience the sublime... Kant's earlier writings show that he was concerned with the problem of motivating the subjective individual under the objectively conceived moral law... Kant recognized that subjective motivation requires something more than an objective principle; this subjective motivation must be a feeling... In any event, it seems safe to say that through the experience of the sublime, which is the experience of our supersensible faculty, one may be motivated to embrace one's innate moral feeling.
Intellectual pleasure comes when one feels one's freedom, and this feeling can only be found in morality. The sublime is a presentation of nature that excites this same intellectual pleasure in us and thereby gives us reason to believe that nature is supportive of the ends of freedom... An interpretation of the feeling of the sublime as phenomenologically important to the problem of moral motivation, since it provides an additional doorway to the supersensible, would seem to be consistent with Kant's apparent belief as evidenced by the lectures and his other published works. Here lies, most probably, the significance of the sublime in Kant's moral metaphysics.
Kant's analysis of the aesthetic judgment presupposes that all rational beings share the same mental faculties. His threefold classification of mental faculties into the cognitive power, the feeling of pleasure and displeasure and the power of desire in both Introductions to the third Critique is precisely a claim about the totality of persons. Kant has in mind the totality of persons generally when describing cognitive psychology, epistemology, moral philosophy or aesthetics. One of his fundamental premises is that "all rational beings" possess certain mental faculties and their given characteristics and limitations.
So Kant argues in the second Critique that through pure practical reason the supersensible is given objective reality. But Kant goes even farther in the third Critique and argues that the aesthetic experience of the sublime allows the mind to feel its supersensible vocation, and here Kant emphasized aesthetics and uses the term supersensible from his famous transcendental idealism to refer to the unity which links the theoretical and moral domains together in order to realize morality's status as the final end of all creation.
In summary Kant's sublime was developed throughout several stages in his career and becomes ultimately for Kant a doorway to the supersensible that is morally significant because it helps to solve the problem of motivating the subject towards an objectively conceived goal.