From WP on transcendental idealism:
Kant means that his philosophical approach to knowledge transcends mere consideration of sensory evidence and requires an understanding of the mind's innate modes of processing that sensory evidence... In the "Transcendental Aesthetic" section of the Critique of Pure Reason, Kant outlines how space and time are pure forms of human intuition contributed by our own faculty of sensibility. Space and time do not have an existence "outside" of us, but are the "subjective" forms of our sensibility and hence the necessary a priori conditions under which the objects we encounter in our experience can appear to us at all. Kant describes time and space as "empirically real" but transcendentally ideal.
Thus, while we apprehend the Ship of Theseus by "empirically real" physical characteristics, it's size, color, weight, etc., the ship exists not merely as a physically real vessel, but a "transcendentally ideal" vessel, and the difference between the transcendentally ideal is not one of mere appearance and actuality. What is actual is noumenon, and cannot be known since knowing entails escaping phenomenon, which is a construction of the mind.
The ship as physical object does not disappear in Kant, it's metaphysical nature just shifts from substance to representation. The paradox of the Ship of Theseus is about how it can remain the same object of cognition while being replaced part by part, ie. about the identity of an object of cognition. You apply a physicalist ontology to Kant here and the compatibility is questionable. – Philip Klöcking
It is certainly questionable for a Continental thinker such as most learned and formally trained sysadmin (and whose knowledge of Kant is far superior to many here) to look for a physically reductionist grounding of Kantian notions, but it is actively a part of the Oxfordian Nicholas Shea's research in his book Representation in Cognitive Science. In such a metaphysical framework of thinking, one might suggest humbly that transcendental idealism does anticipate certain findings such as those which show that perception provides consciousness a construction as per the research programme of visual computation. Thus, what metaphysically presents itself as the Ship of Theseus problem of identity, might be suggested to unravel into a constructivist epistemology in so far as our intuitive notions of the existence of things isn't a denial of the substance involved in identity, that is the objective form of matter and the existence that can be broadly construed as a noumenological ship, but rather furthering Kant's motivations of characterizing identity and categorization within a program of accepting neural computation and conceptual metaphor as building blocks of thoughts in contradistinction to more classical philosophical notions in the philosophy of mind such as the language of thought (SEP). In this way, one rejects the contemporary arguments to re-embrace direct realism such as Searle argues, and at the same time, dispense with the notion that human thought is primarily linguist and rational, but instead is seated in what Searle himself calls "The Background", that nebulous state of affairs most philosophers are happy to call intuition.
Does Kant's philosophy of perception and intuition imply that the unity of perceived individuals is an intuition? If so, this seems to resolve the various paradoxes of physical individuals such as the problem of change over time and the ship of Theseus
On my reading, yes. Kant was attempting to show in transcendental idealism and with his phenomenological and noumenological distinction that subjectivity and objectivity and the mental and the physical are cross-cutting concerns, to use contemporary CS-AOP terminology. Your intuition that Kant anticipates dissolving the paradox inherent in the Ship of Theseus is an accurate one if you accept that conceptualization is a process of the mind that is anchored in linguistic faculty and not some Platonic intuition of a transcendental reality in which concepts exist as real forms. That is to say, that an anti-realist approach to categorization can be seen as a natural outcome of Kantian thinking. This too seems to affirm late LW in PI where he posits family resemblance, an idea taken up and furthered by Eleanor Rosch and eventually George Lakoff starting with prototype theory to understand the linguistic basis of concept and the nature of apriorticity.