I'm assuming this James Akin, who wrote the linked article, is the same well-known apologist Jimmy Akin from Catholic Answers. Jimmy has a tendency, in my experience, to sometimes over-simplify concepts to the point that important nuance is lost, and I believe he does so in some places here.

Let's examine each article of TULIP.

Total Depravity

Akin says the following

Despite its name, the doctrine of total depravity does not mean men are always and only sinful. Calvinists do not think we are as sinful as we possibly could be. They claim our free will has been injured by original sin to the point that, unless God gives us special grace, we cannot free ourselves from sin and choose to serve God in love. We might choose to serve him out of fear, but not out of unselfish love [9].

What would a Catholic think of this teaching? While he would not use the term "total depravity" to describe the doctrine [10], he would actually agree with it. The accepted Catholic teaching is that, because of the fall of Adam, man cannot do anything out of supernatural love unless God gives him special grace to do so [11].

Thomas Aquinas declared that special grace is necessary for man to do any supernaturally good act, to love God, to fulfill God's commandments, to gain eternal life, to prepare for salvation, to rise from sin, to avoid sin, and to persevere [12].

This is not precisely correct. In the frist place, Calvinists do not even totally agree (no pun intended) on what total depravity is. Some Calvinists give the definition Akin uses. Others supply an important caveat. From GotQuestions.org:

What the Bible does teach and what total depravity does recognize is that even the “good” things man does are tainted by sin because they are not done for the glory of God and out of faith in Him (Romans 14:23; Hebrews 11:6). While man looks upon the outward acts and judges them to be good, God looks upon not only the outward acts but also the inward motives that lie behind them, and because they proceed from a heart that is in rebellion against Him and they are not done for His glory, even these good deeds are like “filthy rags” in His sight.

This is not the Catholic position. The Catholic Theologian would recognize that a good act is good even with only naturally good intentions. So, for instance, if an atheistic surgeon were to complete a life saving surgery, not because he is seeking his own glory, but because he wills the good of the patient, namely, for the patient to continue living, then that atheist has done a good thing. No Catholic could agree that the atheistic surgeon sinned in this act, because the object is good and the circumstances, including the dispotition of his heart, are not such that the act could become sinful.

Could the act have been better? Yes, the atheist could have offered his act to the glory of God, and it would have been meritorious for him to do so, but the act is not sinful, or 'tainted by sin,' merely because the surgeon is not a Christian.

Akin's presentation of Aquinas' view, that grace is required for men to perform good supernatural acts, is also misleading. Of course, the keyword here is supernatural. The doctrine of Total Depravity states (depending on the Calvinist you ask) that all good acts are impossible without a special grace, not just supernatural acts.

Unconditional Atonement

Akin appears to present Unconditional Atonement correctly, but the issue is in the common Calvinist inference, as he puts it, of "double predestination." As the article you linked explains, this is entirely incompatible with Catholic teaching. Akin even quotes the Council of Trent. Double predestination is the claim that God actively wills both the salvation of the elect and the damnation of the reprobate. He does not, because then He would not only will an evil thing, but would induce the reprobate to sin.

Limited Atonement

The doctrine of Limited Atonement is either a tautology, if you take it to mean that only the elect will receive salvation, or it must be contrary to Catholic teaching, if you take it to mean that Christ only died for the sake of the elect. Akin quotes Aquinas on this issue: 'Aquinas stated, "Christ's passion was not only a sufficient but a superabundant atonement for the sins of the human race; according to 1 John 2:2, 'He is the propitiation for our sins, and not for ours only, but also for those of the whole world.'"' The TULIP explanation on ligonier isn't very clear on this question, stating:

I prefer not to use the term limited atonement because it is misleading. I rather speak of definite redemption or definite atonement, which communicates that God the Father designed the work of redemption specifically with a view to providing salvation for the elect, and that Christ died for His sheep and laid down His life for those the Father had given to Him.

Does this mean that only the elect will benefit from Christ's sacrifice, or does it mean that Christ only died for the sake of the elect? It is unclear, but it appears to lean towards the latter view, which would put it at odds with the Catholic Church.

Irresistable Grace

The doctrine of Irresistable Grace might be reconcilable with Catholic Theology. The Calvinist puts it thus:

The Westminster Confession (10.1) reminds us that God’s irresistible grace does not save people against their wills but by “renewing their wills . . . so as they come most freely, being made willing by his grace.” Unfortunately, the term irresistible can suggest capricious force or violence to a sinner’s will. If you are a believer, you know that when grace took hold of you, it brought you willingly and lovingly to what God had predetermined for you.

Of course, the grace which renews the will of men saves them. This, again seems a tautology. This is the grace received when one is baptized or confesses their sins: it is sanctifying grace. Akin gives a (surprisingly) nuanced explanation in the linked article:

This is the principal issue between Thomists and Molinists. [29] Thomists claim this enabling grace is intrinsically efficacious; by its very nature, because of the kind of grace it is, it always produces the effect of salvation. Molinists claim God's enabling grace is only sufficient and is made efficacious by man's free choice rather than by the nature of the grace itself. For this reason Molinists say that enabling grace is extrinsically efficacious rather than intrinsically efficacious. [30]

A Catholic can agree with the idea that enabling grace is intrinsically efficacious and, consequently, that all who receive this grace will repent and come to God. Aquinas taught, "God's intention cannot fail... Hence if God intends, while moving it, that the one whose heart he moves should attain to grace, he will infallibly attain to it, according to John 6:45, 'Everyone that has heard and learned from the Father comes to me.'" [31] Catholics must say that, while God may give efficacious grace only to some, he gives sufficient grace to all. This is presupposed by the fact that he intended the atonement to be sufficient for all. Vatican II stated, "[S]ince Christ died for all men, and since the ultimate calling of man is in fact one and divine, we ought to believe that the Holy Spirit in a manner known only to God offers to every man the possibility of being associated with this paschal mystery." [32]

Perseverance of the Saints

This is perhaps the most difficult point in Calvinist theology to reconcile with Catholicism. The final doctrine of TULIP is often colloquially refered to as "once saved, always saved." This clearly contradicts the Catholic teaching that sanctifying grace, that "intrinsically efficacious grace" as Akin puts it near the end of his article, can be lost through mortal sin. Akin puts it thus:

A Catholic must affirm that there are people who experience initial salvation and who do not go on to final salvation, but he is free to hold to a form of perseverance of the saints. The question is how one defines the term "saints"--in the Calvinist way, as all those who ever enter a state of sanctifying grace, or in a more Catholic way, as those who will go on to have their sanctification (their "saintification") completed. [42] If one defines "saint" in the latter sense, a Catholic may believe in perseverance of the saints, since a person predestined to final salvation must by definition persevere to the end. Catholics even have a special name for the grace God gives these people: "the gift of final perseverance."

The doctrines are clearly distinct. The mere fact that one can call the gift of final perseverance "Perseverance of the Saints" does not mean that the actual understanding here is the same. Calvinists understand that anyone who has received God's irresistable and salvific grace will persevere, and Catholics do not believe this at all. In case we would doubt that the Calvinist really means this, our Calvinist source puts it thus:

The word perseverance might be somewhat confusing because it could seem to communicate that God has started something, and now it is your turn: you must persevere. The biblical teaching, however, is that God has done something; God is doing something; and God will do something. The God who starts is the God who finishes.

Conclusion

Akin ends his essay with this statement

There are other ways to construct a Thomist version of TULIP, of course, but the fact there is even one way demonstrates that a Calvinist would not have to repudiate his understanding of predestination and grace to become Catholic. He simply would have to do greater justice to the teaching of Scripture and would have to refine his understanding of perseverance.

In fact the Calvinist need not only refine his understanding of perseverance. He needs to reform it entirely. He must reject what U, L, and P really assert under most Calvinist formulations, and would need to reject the T under many conceptions. Only Irresistable Grace could be salvaged in tact for the Calvinist convert to Catholicism, with a particular understanding of which grace is being dispensed, and at what times, and by which Sacramental means.