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Catholicism does not argue anything like this.
The Catholic method of proving the Bible to be inspired is this: The Bible is initially approached as any other ancient work. It is not, at first, presumed to be inspired. From textual criticism we are able to conclude that we have a text the accuracy of which is more certain than the accuracy of any other ancient work.
Next we take a look at what the Bible, considered merely as a history, tells us, focusing particularly on the New Testament, and more specifically the Gospels. We examine the account contained therein of Jesus’ life, death, and resurrection.
Using what is in the Gospels themselves and what we find in extra-biblical writings from the early centuries, together with what we know of human nature (and what we can otherwise, from natural reason alone, know of divine nature), we conclude that Jesus was who he claimed to be—God.
Further, Christ said he would found a Church. Both the Bible (still taken as merely a historical book, not yet as an inspired one) and other ancient works attest to the fact that Christ established a Church with the rudiments of what we see in the Catholic Church today—papacy, hierarchy, priesthood, sacraments, and teaching authority.
We have thus taken the material and purely historically concluded that Jesus founded the Catholic Church. Because of his Resurrection we have reason to take seriously his claims concerning the Church, including its authority to teach in his name.
This Catholic Church tells us the Bible is inspired, and we can take the Church’s word for it precisely because the Church is infallible. Only after having been told by a properly constituted authority—that is, one established by God to assure us of the truth concerning matters of faith—that the Bible is inspired can we reasonably begin to use it as an inspired book.
Note that this is not a circular argument. We are not basing the inspiration of the Bible on the Church’s infallibility and the Church’s infallibility on the word of an inspired Bible. That indeed would be a circular argument!
What we have is really a spiral argument. On the first level we argue to the reliability of the Bible insofar as it is history. From that we conclude that an infallible Church was founded. And then we take the word of that infallible Church that the Bible is inspired.
This is not a circular argument because the final conclusion (the Bible is inspired) is not simply a restatement of its initial finding (the Bible is historically reliable), and its initial finding (the Bible is historically reliable) is in no way based on the final conclusion (the Bible is inspired).
What we have demonstrated is that (1) the Church gets its authority from Jesus, not the Bible per se, and (2) without the existence of the Church, we could never know whether the Bible is inspired.