> A lot of Catholic morality derives from the postulate that man was specially made by God and "in God's image" which gives man an inherent, unique-among-all-creation dignity
The Church's understanding of morality draws heavily from natural law theory. Natural law theory grounds morality in human nature: what is good for human beings is determined by what it means to be human. Morality enters the picture, because unlike other animals or beings, a central part of what it means to be human is rational and to be able to choose freely between apprehended alternatives. This forms the basis for rights and responsibilities.
Now, it would be a mistake to say that the Imago Dei does not inform this understanding. In fact, the image of God consists of Man's rationality and freedom which stands in analogous relation to God (God is obviously infinitely different from human beings, but nonetheless the analogia entis holds, because it is analogical, neither univocal nor equivocal). It is Man's nature as intellectual being that makes him created in the image of God. (Angels, too, are created in the image of God for the same reasons. They have angelic intellects which differ from human intellects; whereas human beings apprehend reality through the senses from which the intellect then abstracts forms imperfectly, angels can apprehend the forms of things directly.)
I would also say that "postulate" is not the right term, as the Church is not postulating. It accepts this as true.
> Its part of the reason why they were initially anti-heliocentrism (man wasn't at the center of the universe) and anti-evolution (man wasn't specially created) before coming around due to overwhelming evidence.
The Universe we inhabit is, in this greater cosmology, quite lowly in comparison. So even if human beings were to inhabit a spatial center (whatever significance you wish to attach to that), it would be a lowly center. W.r.t. evolution, the opposition the Church has is not to various biological explanations of change and adaptation, but evolutionism, which is a metaphysical position, not a biological one, one that many who advocate for evolution also hold without realizing it is the domain of metaphysics, not biology. The Church still holds that each soul is the result of a special act of creation. I won't get into the metaphysics here, but it is decidedly not Cartesian.
> A lot of Catholic theology regarding the soul is driven by god-of-the-gaps style reasoning.
I have no idea what you mean here. The intellect and will are held to be immaterial faculties, making human souls intellectual [0]. Aristotle gives arguments for this position. Roughly, the intellect cannot be a purely physical faculty, because abstraction ultimately involves the separation of form from particulars. Because matter (understood as prime matter, etc) is the particularizing principle, the joining of matter with form is what is the cause of concrete instances of that form. Thus, if the intellect were material, the apprehension of form would mean the instantiation of the apprehended thing in the intellect as a particular, which is clearly not what happens! When you apprehend "triangularity" or "Horseness", you do not instantiate a concrete triangle or a concrete horse in your mind! And, in fact, if you did, you would by the very act fail to grasp the universal concept, because particulars by definition exclude all others particulars except themselves. You would possess this triangle or this horse, and not any other of the potentially infinite instances of them. You would not grasp what it means to be a triangle or a horse.
So, it is not a matter of the Church feeling threatened in some way. Concerns have nothing to do with some kind of conceptual threat to the "specialness" of human beings. AI, on this account, simply cannot reason; if it could, then it could, but it cannot. The computational formalism is, to put it in Searlian terms, all syntax and no semantics, which is to say no intentionality. And even here, the physical device isn't even objectively a computer and isn't objectively computing (both Searle and Kripke present arguments for this, for example). But whether computers can reason is actually besides the point.
> Obviously no one today says that the soul is in charge of the "locomotion" of living things
You seem to misunderstand what a soul is. The soul is the form of a living thing. Thus, the soul of horse is that principle which causes it to be the kind of thing it is, and thus is its organizing principle. This isn't Cartesian metaphysics here where you have one thing, the res cogitans, and a second thing, the res extensa, kind of glued to one another, but really two separate things. By analogy, if you have a sphere of bronze, then the "soul" of that ball is the "sphericity". The sphericity makes the ball of bronze what it is. The sphere ceases to be a sphere if you were to melt it or hammer it into a cube.
If this topic interests you, you will find Feser's "Immortal Souls" interesting [1]. He gives a thorough treatment of the subject.
[0] https://www.youtube.com/watch?v=w6GmCyKylTw
[1] https://a.co/d/6fWau6Q