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...