I will give you a two-fold answer.
One, the Church does not find what St. Paul preached to be at odds with th Gospels. The Church interpretation is in harmony with the Gospels. +Paul is interpretd through the Gospels, as the Old testament is interpreted through the New Testament. Everything is harmonized. But this hamrony did not come about naturally.
Two, historically, +Peter +James, on one side, and +Paul on the other, very very much at odss with each other. The book of Acts (written after both Apostles were dead) tries to smooth things over in that respect. Even later than that, the book of 1 Peter appeared. Written towards the end of the century, it tries to mend the rift between the followers of +Peter and those of +Paul.
[Galatians 2:9]
“The book of Acts (written after both Apostles were dead) tries to smooth things over in that respect. Even later than that, the book of 1 Peter appeared. Written towards the end of the century, it tries to mend the rift between the followers of +Peter and those of +Paul.”
That was not the position of scholars until the Tubingen, Higher Criticism, crowd.
From F.F. Bruce, “The New Testament Documents: Are They Reliable?”
“The New Testament was complete, or substantially complete, about AD 100, the majority of the writings being in existence twenty to forty years before this. In this country a majority of modern scholars fix the dates of the four Gospels as follows: Matthew, c. 85-90; Mark, c. 65; Luke, c. 80-85; John, c. 90-100.4 I should be inclined to date the first three Gospels rather earlier: Mark shortly after AD 60, Luke between 60 and 70, and Matthew shortly after 70. One criterion which has special weight with me is the relation which these writings appear to bear to the destruction of the city and temple of Jerusalem by the Romans in AD 70. My view of the matter is that Mark and Luke were written before this event, and Matthew not long afterwards.
But even with the later dates, the situation’ encouraging from the historian’s point of view, for the first three Gospels were written at a time when man, were alive who could remember the things that Jesus said and did, and some at least would still be alive when the fourth Gospel was written. If it could be determined that the writers of the Gospels used sources of information belonging to an earlier date, then the situation would be still more encouraging. .
The date of the writing of Acts will depend on the date we affix to the third Gospel, for both are parts of one historical work, and the second part appears to have been written soon after the first. There are strong arguments for dating the twofold work not long after Paul’s two years’ detention in Rome (AD 60-62)Some scholars, however, consider that the ‘former treatise’ to which Acts originally formed the sequel was not our present Gospel of Luke but an earlier draft, sometimes called ‘ProtoLuke’; this enables them to date Acts in the sixties, while holding that the Gospel of Luke in its final form was rather later.”
from Guthrie, “New Testament Introduction” 1st Peter,
“So strong is the evidence for the use of this epistle in the early church that some scholars have regarded it as proved and maintained that it was considered to be canonical as early as this word had a meaning.” There are parallels in Clement of Rome’s Epistle to the Corinthians, Ignatius, Barnabas, and Shepherd of Hermas. These may indicate borrowing, but not necessarily. Polycarp definitely quotes from it, though he does not identify the quoted material as coming from Peter. Irenaeus, however, does quote from it, and regards it as a genuine work of Peter. From the last third of the second century on, this letter is frequently regarded as Petrine, and is cited by Tertullian, Clement of Alexandria, Theophilus of Antioch, etc.
Although it may not have been used as freely in the West as in the East, there is no evidence that it was ever disputed.
…the primitive church, as far back as any evidence exists, regarded it as a genuine epistle of Peter, and thus any discussion of objections to Petrine authorship must sufficiently take account of this fact.”