The High Priest Joseph Caiaphas was a puppet priest, put into office by the Romans. Caiaphas did not have the power of speaking on the level of infallibility as does the Pope when speaking on faith and morals.
References in the Mosaic Law to "the death of the high priest" (Num 35:25, 28) suggest that the high-priesthood was ordinarily held for life. Perhaps for this reason, Annas was still called "high priest" even after his dismissal, along with Caiaphas (Luke 3:2). Annas officially served as High Priest for ten years (6–15 A.D.), when at the age of 36 he was deposed by the procurator Gratus. Yet while having been officially removed from office, he remained as one of the nation's most influential political and social individuals, aided greatly by the use of his five sons and his son-in-law Caiaphas as puppet High Priests.3 - Annas (Wikipedia)
In your question you state that Caiaphas, as high priest, declares an infallible statement by the power of the chair of Moses. Yet you do not support your claims. Yes, he was High Priest that year, but named so by Roman authority, not Divine.
The use of infallibility in Catholic theology is rather restrictive; it is a Catholic dogma and has been taught through the ages. It means that, by Divine assistance, the Church (of Christ), is preserved from error, or even the possibility of error, in definitive dogmatic teachings regarding faith and morals. This does not mean that individual believers are infallible in their subjective interpretations and it does not require a holiness of life after all, Caiaphas was given the greater gift of prophecy when he condemned Christ. Surely a lesser gift would not require more. Christ intended every person to be part of His Church and that the Church be one in faith and worship. In doing so, He established a visible Church and gave authority to the apostles and their successors, exclusively, in order to teach and govern.
Infallibility differs from inspiration in that inspiration is a special Divine influence that not only preserves from error (inerrant) but controls what the subject can say or write as the Word of God. Inspiration directly makes God the author of inspired utterances whereas infallibility only preserves the subject from error. In the case of infallibility, God is not the author.
The Catholic Encyclopedia has this to say about Joseph Caiaphas:
According to Josephus (Antiquitates, XVIII, iv, 3), Caiphas was appointed High-Priest of the Jews by the Roman procurator Valerius Gratus, the predecessor of Pontius Pilate, about A.D. 18 (Ant., XVIII, ii, 2), and removed from that office by the procurator Vitellius, shortly after he took charge of affairs in Palestine, A.D. 36 (Ant., XVIII, iv, 3). During this period the famous Annas, father-in-law of Caiphas (John 18:13), who had been high-priest from A.D. 6 to 15, continued to exercise a controlling influence over Jewish affairs, as he did when his own sons held the position. This explains the rather puzzling expression of Luke 3:2, epi archiereos Anna kai Kaiapha (under the high-priest Annas and Caiphas; cf. Acts 4:6). Caiphas was certainly the only official high-priest at the time St. Luke refers to, at the beginning of the public life of Christ; but Annas still had his former title and a good deal of his former authority. The role assigned him in the trial of Christ, in John 18, points to the same continued influence. In the measures taken by the Jewish authorities to do away with Jesus, Caiphas certainly had the most discreditable part. After the raising of Lazarus, the priests and Pharisees held council to determine what was to be done in view of the manifest signs of the Prophet of Nazarus and what they were pleased to consider the danger resulting to the country. The words of Caiphas, the high-priest of that year, are reported by St. John: "You know nothing. Neither do you consider that it is expedient to you that one man should die for the people, and that the whole nation perish not" (11:49-50). They show a disdain for others, and a determination to get rid of this man who was displeasing to him, without any consideration of the justice of his cause. But while we may see in the declaration of Caiphas the manifestation of very unworthy sentiments, we are warned by St. John that it was prophetical. The high-priest expressed in a striking way the meaning of the sufferings of the Man-God (John 11:51-52), though he could not have realized the full import of those mysterious words. The death of Jesus being resolved upon, the most unscrupulous means were employed in order to bring it about, and Caiphas is chiefly to blame. The meeting determined upon by the princes of the priests and the elders of the people, "that by subtlety they might apprehend Jesus", was held in the house of Caiphas (Matthew 26:3-5). The hill south of Jerusalem where this house is said by tradition to have stood is called the "Hill of Evil Counsel". As high-priest, Caiphas was the official head of the Sanhedrin, and consequently responsible for the travesty of a trial to which Christ was submitted by the Jewish authorities, before they handed Him over to Pilate and stirred up the people to demand his death.
After the death of Jesus, Caiphas continued to persecute his followers. When Peter and John were brought before the Council after the cure of the lame man at the Beautiful Gate of the Temple (Acts 4:6 sqq.), Caiphas was still high-priest, since he was removed A.D. 36 or 37. We can say with almost equal certainty that he was the high-priest before whom St. Stephen appeared (Acts 7:1), and that it is from him that Saul obtained letters authorizing him to bring the Christians of Damascus to Jerusalem (Acts 9:1-2). At a time when high-priests were made and unmade by officials of Rome, and when the principal quality required seems to have been subserviency, it is no credit to the character of Caiphas to have enjoyed their favour so long. Josephus mentions his rule in connection with a series of acts of Vitellius which were agreeable to the Jews. We are not told what became of him after his deposition.
The power of binding and loosening was claimed by the Pharisees, but not in the exact sense that the Catholic Church employs the term. The following is from the Jewish Encyclopedia!
The power of binding and loosing was always claimed by the Pharisees. Under Queen Alexandra, the Pharisees, says Josephus ("B J." i, 5, § 2), "became the administrators of all public affairs so as to be empowered to banish and readmit whom they pleased, as well as to loose and to bind." This does not mean that, as the learned men, they merely decided what, according to the Law, was forbidden or allowed, but that they possessed and exercised the power of tying or untying a thing by the spell of their divine authority, just as they could, by the power vested in them, pronounce and revoke an anathema upon a person. The various schools had the power "to bind and to loose"; that is, to forbid and to permit (Ḥag. 3b); and they could bind any day by declaring it a fast-day (Meg. Ta'an. xxii.; Ta'an. 12a; Yer. Ned. i. 36c, d). This power and authority, vested in the rabbinical body of each age or in the Sanhedrin (see Authority), received its ratification and final sanction from the celestial court of justice (Sifra, Emor, ix.; Mak. 23b).
In the New Testament.
In this sense Jesus, when appointing his disciples to be his successors, used the familiar formula (Matt. xvi. 19, xviii. 18). By these words he virtually invested them with the same authority as that which he found belonging to the scribes and Pharisees who "bind heavy burdens and lay them on men's shoulders, but will not move them with one of their fingers"; that is, "loose them," as they have the power to do (Matt. xxiii. 2-4). In the same sense, in the second epistle of Clement to James II. ("Clementine Homilies," Introduction), Peter is represented as having appointed Clement as his successor, saying: "I communicate to him the power of binding and loosing so that, with respect to everything which he shall ordain in the earth, it shall be decreed in the heavens; for he shall bind what ought to be bound and loose what ought to be loosed as knowing the rule of the church." Quite different from this Judaic and ancient view of the apostolic power of binding and loosing is the one expressed in John xx. 23, where Jesus is represented as having said to his disciples after they had received the Holy Spirit: "Whosesoever sins ye remit, they are remitted unto them; and whosesoever sins ye retain, they are retained." It is this view which, adopted by Tertullian and all the church fathers, invested the head of the Christian Church with the power to forgive sins, the "clavis ordinis," "the key-power of the Church." - BINDING AND LOOSING
But what was the Seat of Moses?
The chair of Moses was a phrase used by Jesus in Matthew 23:2 to signify the place of authority that the Scribes and Pharisees had in interpreting The Law and exercising their authority over the Jewish people. They are the ones who would tell the people of Israel what the law of Moses "really" meant. The phrase is found only in this verse. In Greek, the phrase is Μωσέως καθέδρας (Moses kathedras), and is literally "Moses' seat." It is translated as "Moses' seat" in the ASV, KJV, ESV, NET, NIV. - What is the Chair of Moses (the Seat of Moses)?
The phrase of sitting on the seat of Moses seems to be more in line with a bishops authority within his own diocese than that of a pope's authority when declaring something Ex Cathedra.
“The scribes and the Pharisees sit on Moses’ seat” (Mt. 23.2), a carved stone chair reserved for a visitor or for the most distinguished elder of a synagogue. This one is from Chorazin, one of the three towns Jesus inveighed against for their citizens’ hypocrisy.
In 1926 a unique stone seat was found near the southern wall of the Chorazin synagogue. Since then it has been called the “Chair of Moses.” The Chair of Moses is a special seat that is used in some synagogues, even today, on certain occasions, usually located near the most important wall, that which faces Jerusalem.
On the Chair of Moses was inscribed in Aramaic- “Remember for good Yudan the son of Ishmael who made this stvh and its steps; may he take part with the pious.” Stvh has been understood as the Greek word stoa (v [vov] and h [he] are so-called matres lectiones, or consonants used as vowels. The Hebrew letter vov was used for a long o and the Hebrew letter he for a short a). A stoa is a portico supported by columns. Another interpretation of this work was published by Jacob N. Epstein in 1930 in a paper on the inscription on the Chair of Moses. According to his view, and ours, stvh refers to the Hebrew or Aramaic word istava which means a shelf, a platform or a small roofed and decorated dais (bema, plural bemot) on which the chair probably rested and from which the scriptures were read or chanted. A small replica of this bema was found on a frieze fragment. - Moses' Seat
Caiaphas did not exercise clerical infallibility at all, in the same way a pope exercises papal infallibility. He was a puppet High Priest under Roman authority. Your question confuses clerical infallibility with the Jewish authority (binding and loosing) of the Scribes, the Pharisees and the High priests who held office at that moment.