What is the dominant theory of atonement in Catholic theology?

There is no single theory of atonement in Catholicism, but rather the doctrine has developed over time due to speculation from various theologians. Though out of the four you've provided, satisfaction theory is the only one I think you will find an orthodox catholic espousing. The Catholic Encyclopedia entry on atonement is helpful.

it may be noted that in this instance the development is chiefly due to Catholic speculation on the mystery, and not, as in the case of other doctrines, to controversy with heretics. At first we have the central fact made known in the Apostolic preaching, that mankind was fallen and was raised up and redeemed from sin by the blood of Christ. But it remained for the pious speculation of Fathers and theologians to enter into the meaning of this great truth, to inquire into the state of fallen man, and to ask how Christ accomplished His work of Redemption (emphasis mine). By whatever names or figures it may be described, that work is the reversal of the Fall, the blotting out of sin, the deliverance from bondage, the reconciliation of mankind with God. And it is brought to pass by the Incarnation, by the life, the sufferings, and the death of the Divine Redeemer. All this may be summed up in the word Atonement. This, is so to say, the starting point. And herein all are indeed at one. But, when it was attempted to give a more precise account of the nature of the Redemption and the manner of its accomplishment, theological speculation took different courses, some of which were suggested by the various names and figures under which this ineffable mystery is adumbrated in Holy Scripture. Without pretending to give a full history of the discussions, we may briefly indicate some of the main lines on which the doctrine was developed, and touch on the more important theories put forward in explanation of the Atonement.

The page explains that substitutionary theories of atonement (and here I include as well penal substitution) are a product of post-reformation protestantism. So we don't follow those theories because our theologians never put them forward.

And in their explanation of the merit of Christ's sufferings and death we may see the influence of St. Thomas and the other great Schoolmen. But, as might be expected from the isolation of the doctrine and the loss of other portions of Catholic teaching, the truth thus preserved was sometimes insensibly obscured or distorted. It will be enough to note here the presence of two mistaken tendencies.

• The first is indicated in the above words of Pattison in which the Atonement is specially connected with the thought of the wrath of God. It is true of course that sin incurs the anger of the Just Judge, and that this is averted when the debt due to Divine Justice is paid by satisfaction. But it must not be thought that God is only moved to mercy and reconciled to us as a result of this satisfaction. This false conception of the Reconciliation is expressly rejected by St. Augustine (In Joannem, Tract. cx, section 6). God's merciful love is the cause, not the result of that satisfaction.

• The second mistake is the tendency to treat the Passion of Christ as being literally a case of vicarious punishment (emphasis mine). This is at best a distorted view of the truth that His Atoning Sacrifice took the place of our punishment, and that He took upon Himself the sufferings and death that were due to our sins.

Any sort of ransom theory also seems unreasonable and has been met with unanswerable arguments from Anselm and others:

But however useful and illuminating in their proper place, figures of this kind are perilous in the hands of those who press them too far, and forget that they are figures. This is what happened here. When a captive is ransomed the price is naturally paid to the conqueror by whom he is held in bondage. Hence, if this figure were taken and interpreted literally in all its details, it would seem that the price of man's ransom must be paid to Satan. The notion is certainly startling, if not revolting. Even if brave reasons pointed in this direction, we might well shrink from drawing the conclusion. And this is in fact so far from being the case that it seems hard to find any rational explanation of such a payment, or any right on which it could be founded... This curious notion, apparently first mooted by St. Irenæus, was taken up by Origen in the next century, and for about a thousand years it played a conspicuous part in the history of theology... A protest was raised by St. Gregory of Nazianzus in the fourth century, as might be expected from that most accurate of the patristic theologians. But it was not till St. Anselm and Abelard had met it with unanswerable arguments that its power was finally broken. It makes a belated appearance in the pages of Peter Lombard.

This is great question and an area where most Catholics would struggle to answer. Many, in fact, would probably end up articulating theories of atonement that are generally more Protestant in nature simply because of their prevalence in society.

The short answer is this: while it is absolutely a dogmatic teaching that Christ's death on the cross makes possible our salvation (because he said so himself at the Last Supper), how his death achieves such an end is a matter of theological investigation and there isn't a single, obvious, "dominant" Catholic theory.

Yes, you can always look up any particular theologian's ideas on the topic—and someone like Aquinas is an obvious example if you're approaching the subject from a Catholic perspective. But I believe your question is looking for something more: why, theologically, would any one atonement theory be more Catholic than another?

To answer that, we have to recognize the tremendously systematic nature of the Catholic theological tradition. In other words, every part of Catholic theology is integrally connected to the other parts. So an atonement theory that is most Catholic in nature would be one that is most compatible with a Catholic understanding of grace, salvation, humanity, and that particularly sacramental worldview so characteristic of Catholicism.

Let's look a just a few of those areas, though I'm sure it would be possible to write a entire volume on the topic (indeed, there probably is one).

First, consider a Catholic understanding of human nature and grace in contrast with Calvinism. For Calvin, humans are totally depraved and ultimately incapable of any goodness in this life. Even acts that, on the surface, appear to be altruistic are selfish and self-serving. God's grace doesn't change that human reality for Calvin—which is why no one could ever "merit" salvation in any way and salvation is completely the work of God alone.

Catholics, on the other hand, see God's grace as sanctifying and transformative. Humans must cooperate with God's grace through free acceptance of it, and in doing so humans can actually reflect God's goodness in this life. That is why Catholics continue to celebrate saints and call everyone to a life of holiness. In short, Catholics believe that God's grace can actually make us—Christ's Body, the Church—a sacramental sign of his love and goodness. Humans aren't always perfect, for sure. But through God's grace, we can be good and loving in this life. Calvin would never say that.

How might that affect one's theology of the cross and salvation? Well, for starters, it is easy to see why Calvin approaches the cross simply as an event in which our punishment gets taken up by Christ. For him, the cross allows us—who are utterly unworthy of salvation—to attain it in a type of theological loophole. A Catholic approach to the cross is going to be much more open to the cross being an event in which Christ's act of love on the cross definitively defeats our slavery to sin and enables us to unite with God in salvation—a process that begins in this life and reaches ultimate fulfillment in heaven. In other words, the cross is an act of love first and foremost. St. Francis de Sales has a lovely Good Friday sermon that speaks well to this approach. But I also think that Karl Rahner, one of the greatest Catholic theologians of the 20th century, also takes an approach in this direction.

I hope this helps...