“In 1 Cor. 4:15 Paul writes, ‘I became your father in Christ Jesus.’ Oops!”
“For though ye have ten thousand instructors in Christ, yet have ye not many fathers: for in Christ Jesus I have begotten you through the Gospel.” (KJB)
“Father” is specific, then to the one who had led them (”begotten” them) to Jesus Christ through the Gospel. There is no reference to a religious order or position, nor to any priesthood. And Paul’s sons in the faith would have been wrong to call him “Father” in the sense of priesthood. They would have been disobeying Christ to do so, and Paul would have known that.
St. Paul, as a father of the Catholic Church, would have been perfectly comfortable with the title "Father," as 1Co 4:15 demonstrates.