Catholic doctrine declare the bible to be the inspired inerrant word of God. I agree. If this is so God breathed the bible into existence. Where we disagree is your claim that the CC cannot err in doctrine. Were did you get such a claim?
The claim comes from Our Apostolic Traditions handed down to us from the successors of the Apostles.
“And thus preaching through countries and cities, they appointed the first-fruits [of their labours], having first proved them by the Spirit, to be bishops and deacons of those who should afterwards believe. Nor was this any new thing, since indeed many ages before it was written concerning bishops and deacons. For thus saith the Scripture a certain place, ‘I will appoint their bishops s in righteousness, and their deacons in faith.’… Our apostles also knew, through our Lord Jesus Christ, and there would be strife on account of the office of the episcopate. For this reason, therefore, inasmuch as they had obtained a perfect fore-knowledge of this, they appointed those [ministers] already mentioned, and afterwards gave instructions, that when these should fall asleep, other approved men should succeed them in their ministry…For our sin will not be small, if we eject from the episcopate those who have blamelessly and holily fulfilled its duties.” Pope Clement, Epistle to Corinthians, 42, 44 (A.D. 98).
“For what is the bishop but one who beyond all others possesses all power and authority, so far as it is possible for a man to possess it, who according to his ability has been made an imitator of the Christ off God? And what is the presbytery but a sacred assembly, the counselors and assessors of the bishop? And what are the deacons but imitators of the angelic powers, fulfilling a pure and blameless ministry unto him, as…Anencletus and Clement to Peter?” Ignatius, To the Trallians, 7 (A.D. 110).
“Hegesippus in the five books of Memoirs which have come down to us has left a most complete record of his own views. In them he states that on a journey to Rome he met a great many bishops, and that he received the same doctrine from all. It is fitting to hear what he says after making some remarks about the epistle of Clement to the Corinthians. His words are as follows: ‘And the church of Corinth continued in the true faith until Primus was bishop in Corinth. I conversed with them on my way to Rome, and abode with the Corinthians many days, during which we were mutually refreshed in the true doctrine. And when I had come to Rome I remained a there until Anicetus, whose deacon was Eleutherus. And Anicetus was succeeded by Soter, and he by Eleutherus. In every succession, and in every city that is held which is preached by the law and the prophets and the Lord.’” Hegesippus, Memoirs, fragment in Eusebius Ecclesiatical History, 4:22 (A.D. 180).
“True knowledge is [that which consists in] the doctrine of the apostles, and the ancient constitution of the Church throughout all the world, and the distinctive manifestation of the body of Christ according to the successions of the bishops, by which they have handed down that Church which exists in every place, and has come even unto us, being guarded and preserved without any forging of Scriptures, by a very complete system of doctrine, and neither receiving addition nor [suffering] curtailment [in the truths which she believes]; and [it consists in] reading [the word of God] without falsification, and a lawful and diligent exposition in harmony with the Scriptures, both without danger and without blasphemy; and [above all, it consists in] the pre-eminent gift of love, which is more precious than knowledge, more glorious than prophecy, and which excels all the other gifts [of God].” Irenaeus, Against Heresies, 4:33:8 (A.D. 180).
“But if there be any (heresies) which are bold enough to plant themselves in the midst Of the apostolic age, that they may thereby seem to have been handed down by the apostles, because they existed in the time of the apostles, we can say: Let them produce the original records of their churches; let them unfold the roll of their bishops, running down in due succession from the beginning in such a manner that [that first bishop of theirs] bishop shall be able to show for his ordainer and predecessor some one of the apostles or of apostolic men,–a man, moreover, who continued steadfast with the apostles. …To this test, therefore will they be submitted for proof by those churches, who, although they derive not their founder from apostles or apostolic men (as being of much later date, for they are in fact being founded daily), yet, since they agree in the same faith, they are accounted as not less apostolic because they are akin in doctrine…Then let all the heresies, when challenged to these two tests by our apostolic church, offer their proof of how they deem themselves to be apostolic. But in truth they neither are so, nor are they able to prove themselves to be what they are not. Nor are they admitted to peaceful relations and communion by such churches as are in any way connected with apostles, inasmuch as they are in no sense themselves apostolic because of their diversity as to the mysteries of the faith.” Tertullian, Prescription against the Heretics, 33 (A.D. 200).
“And that you may still be more confident, that repenting thus truly there remains for you a sure hope of salvation, listen to a tale? Which is not a tale but a narrative, handed down and committed to the custody of memory, about the Apostle John. For when, on the tyrant’s death, he returned to Ephesus from the isle of Patmos, he went away, being invited, to the contiguous territories of the nations, here to appoint bishops, there to set in order whole Churches, there to ordain such as were marked out by the Spirit.” Clement of Alexandria, Who is the rich man that shall be save?, 42 (A.D. 210).