"Mesopotamia" is a place. "Babylon" is a city. There was a "Babylonian Empire," but that had ceased to exist long before. The Epistle says "in Babylon," not in "Babylonia," or "Mesopotamia," or "in the Babylonian empire".
And by the way, the Iraqi Church might not have a tradition of having been founded by the apostle Peter
They have no tradition of Peter having been there at all.
On the other hand, we definitely know that mid-first century Jewish apocalyptic literature referred to Rome as "Babylon," and we know that the Roman Christians have a very ancient tradition of Peter having been there, attested to by Irenaeus, Eusebius, and others, and also attested to by the archaeological evidence found under St. Peter's basilica.
Similarly, we know from tradition that Peter was in Antioch before he was in Rome.
Are you still citing those apocryphal sources as authoritative.
So then are you saying that the "Babylonian Talmud" was really written in Rome? That it is really a code word for the "Roman Talmud"?
The mystics may not have known where Babylon was, or that it was the place of their diaspora, but the Jews of Peter's day sure did. He was writing to the Jews of Asia Minor, and as such used the name of the place with which they were familiar: Babylon or Babylonia, they knew what he meant and where he was.
Do they have Peter's first epistle in their Bibles.? Do they have verse 5:13 of that epistle in their Bibles? If they do, then they have that "tradition", but must have just forgotten about it when the Roman magisterium came to town.