Historical evidence shows early Christianity split between Peter and Paul. The two didn't get along theologically. I doubt that +Peter learned any Greek from +Paul. +Paul was in present-day Turkey and Greece. Peter was nowhere near.
The Jews living in Asia Minor and Greece when +Paul was there were not exiles. There were exiles when wide-spread anti-Christian persecutions began at the end of the 1st century. The Cappadocian Christians dug up a whole city in the mountains where they were hiding.
The Greek text of 1 Peter uses the word διασπορᾶς (diasporas), from which the English word "diaspora" is derived, which means scattering, exile. It is the same word used in the Greek Bible for the Babylonian exile of the Jews.
Surely you don't mean to imply that the Jews who left Israel 500 BC were still "exiles" in Alexandria and Cappadocia! The new exiles 1 Peter speaks of are those contemporary of the author's time.
And when +Peter was alive (c 65 AD) there was no exile#151;yet! The Temple was to stand another 5 years at the time of +Peter's martyrdom. The Jews began to leave Palestine after the destruction of the Temple (70 AD).
But, I agree with Kolo, it doesn't matter to me who wrote it. 1 Pet was chosen by the Church as scripture. That means that in it is God's message, according to the teaching of Christ.
Kolo says (#5058) The Greek [of 1 Peter] is beautiful. That alone makes me doubt that the Apostle Peter wrote it. Add to it Kosta's idea that the author is writing to people fleeing a persecution and it seems even less likely that +Peter is the author. Of course, to me that makes little difference
Indeed, it is ranked higher than that of +Paul's koine Greek! But, as you say, it makes very little difference if +Peter wrote it or not. The Church found it to be scripture because it contains God's message according to the teaching of our Lord.
Well, if true, that would pretty much blow out of the water any criticism of differing Protestant denominations. Right? :) I mention it because learned Catholics have strenuously argued to me that this was NOT in fact the case. They have argued to me that Peter, Paul, and even James were all on the same page, but maybe with different approaches. I happen to agree with the Catholics here. :)