The Greeks called the area of Babylon "Mesopotamia", but the Jews called the area of Mesopotamia "Babylon". And if you notice on the day of Pentecost in Jerusalem, there are Jews there in Jerusalem from Mesopotamia. "Babylon" had been an empire not just a city. It was Mesopotamia as well --- the land between the rivers and there were a lot of people there.
And by the way, the Iraqi Church might not have a tradition of having been founded by the apostle Peter [no one said they did, especially if it was founded shortly after 30 AD by those Jews there in Jerusalem on the day of Pentecost from Mesopotamia], but they do have a copy of the letter postmarked from there by the apostle Peter in their Bibles, don't they?
"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.