It seemed to me you were confused as to Mary's role as Mediatrix. I guess you were not.
In any event, you must beleive that, at least when she wrote and compiled the Bible, Holy Mother Church was infallible.
Now, the difficulty arises in advancing an explanation as to how that once infallible Church failed that doesn't render nugatory the promises of Jesus about His Church He established.
First, the Holy Mother Church's history is a bit questionable (such as Peter's role as the first Pope). During the time that Scripture was written, the church was based not in Rome but in Jerusalem as is evident by the Council at Jerusalem. James was apparently lead there and Peter at some point was rebuked by Paul.
Scripture was penned under inspiration of the Holy Spirit. but NOWHERE in Scripture does it say that the Church itself is infallible. Paul, in fact, commends the Bereans for daring to question him.
So, as they wrote Scripture under the Holy Spirit's guidance, yes, they were writing infallibly. As a church body making decisions and sharing (sometimes contradictory) decisions, they were not.
The Rock that Jesus was to build His church upon was not Peter. Rather, it was Peter's confession or the truth that Jesus Christ is the Lord, the Messiah, God in the flesh.