According to Catholicism, did Judas believe and profess that Jesus is the Son of Man?
The short answer is yes and no.
Judas believed that Jesus was the Son of Man, but did not profess such a belief, at least not publicly or according to the Gospels?
The Son of man indeed goeth, as it is written of him: but woe to that man by whom the Son of man shall be betrayed: it were better for him, if that man had not been born. Matthew 26:24 DRA
In the Gospels, Jesus noted that he was the Son of Man. A point that Judas Iscariot never denied. Judas remained an Apostle of Christ, even after hearing some of the more harder subject matters to digest that Jesus had preached to the people.
True that many thought Jesus was simply a rabbi, but his intimate disciples truly understood that Jesus was indeed the Son of Man.
Jesus Deepens the Scandal: Eat My Flesh, Drink My Blood (6:52-59)
When Jesus mentions his flesh, the tension in the crowd increases. The people are not just grumbling (v. 41); they are arguing sharply with one another (v. 52). Once again we see people who come to Jesus as a rabbi, who even wanted to make him king, but who are far from treating him as either a king or a rabbi. They are not receiving his teaching, as cryptic and offensive as it is. Like Nicodemus, they can only ask how such a thing can be (v. 52; cf. 3:9). "When questioning concerning the `how' comes in, there comes in with it unbelief" (Chrysostom In John 46.2). And Jesus does not make it easy for them. He now makes sure they get the point that real eating and drinking are involved. As he deepens the offense in these verses, he also explains in a very profound way the eternal life he is offering.
Jesus begins by revealing more sharply our need of the life he offers: I tell you the truth, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you (v. 53). He claims that the life he is talking about is not merely some optional gift that we can afford to ignore. Apart from the life he offers, we are dead. Here is a claim as demanding as are his earlier claims about his own identity and what he offers to those who believe in him (vv. 30-51). Our utter neediness is seen clearly when set against the greatness of his offer: Whoever eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day (v. 54). Jesus is promising a new quality of life now and resurrection in the future.
Judas remained the midst of all of Jesus’ preaching and along with all the other disciples stayed at his side, for Christ was obviously the Son of Man!
The most chilling question proposed to Judas by Christ was not whether he denied he was the Son of Man, but rather was he betraying his Master with a kiss?
St. John likewise lays stress on the instigation of the evil spirit: "the devil having now put into the heart of Judas Iscariot, the son of Simon, to betray him" (13:2). The same Evangelist, as we have seen, tells of an earlier intimation of Christ's foreknowledge of the betrayal (John 6:71-2), and in the same chapter says expressly: "For Jesus knew from the beginning, who they were that did not believe, and who he was, that would betray him" (6:65). But he agrees with the Synoptics in recording a more explicit prediction of the treachery at the Last Supper: "When Jesus had said these things, he was troubled in spirit; and he testified, and said: Amen, amen I say to you, one of you shall betray me" (John 12:21). And when St. John himself, at Peter's request, asked who this was, "Jesus answered: He it is to whom I shall reach bread dipped. And when he had dipped the bread, he gave it to Judas Iscariot, the son of Simon. And after the morsel, Satan entered into him. And Jesus said to him: That which thou dost, do quickly. Now no man at the table knew to what purpose he said this unto him. For some thought, because Judas had the purse, that Jesus said to him: Buy those things which we have need of for the festival day: or that he should give something to the poor" (12:26-9). These last details about the words of Jesus, and the natural surmise of the disciples, are given only by St. John. But the prediction and the questioning of the disciples are recorded by all the Synoptics (Matthew 26; Mark 14; Luke 22). St. Matthew adds that Judas himself asked, "Is it I, Rabbi?" and was answered: "Thou hast said it" (26:25).
All four Evangelists agree in regard to the main facts of the actual betrayal which followed so closely on this prediction, and tell how the traitor came with a multitude or a band of soldiers from the chief priests, and brought them to the place where, as he knew, Jesus would be found with His faithful disciples (Matthew 26:47; Mark 14:43; Luke 22:47; John 18:3). But some have details not found in the other narratives. That the traitor gave a kiss as a sign is mentioned by all the Synoptics, but not by St. John, who in his turn is alone in telling us that those who came to take Jesus fell backward to the ground as He answered "I am he." Again, St. Mark tells that Judas said "Hail, Rabbi" before kissing his Master, but does not give any reply. St. Matthew, after recording these words and the traitor's kiss, adds: "And Jesus said to him: Friend, whereto art thou come:" (26:50). St. Luke (22:48) gives the words: "Judas, dost thou betray the Son of man with a kiss?"
Judas Iscariot (Catholic Encyclopaedia)
Judas knew who Jesus was.