My answer would be, yes, this is a Godly prayer, for the following reasons:
1) Rev 6:10-11 pictures martyred saints under the altar of God, in heaven asking how long before their blood is avenged. The response is that they are given white robes, and told to hold on a little longer until their number is fulfilled. Since these are saints, in heaven, under God's altar, we can only conclude that their prayer is a righteous and Godly prayer. Is there truly a difference between their prayer for vengeance and the prayer against Babylon in Psalm 137?
2) Jeremiah chapter 50 (the whole chapter) is just one example of God's Judgment being poured out specifically against Babylon.
3) Revelation 14:8 and 16:19 are two more examples of God's Judgment specifically on Babylon. As this book was written at the very least many centuries after Jeremiah's, we can see proof that God changes not, and that his Word stands eternal. The iniquity of Babylon is not forgotten.
4) While we, as Christians, benefit undeservedly from God's Love, Mercy, and Grace, we cannot forget that God's Word (which is Christ, for the Word was made flesh) is a two-edged sword, healing and restoring with one strike, and cutting to pieces with another stroke. Note John the Baptists description of Christ in Mat. 3:12 - "Whose fan is in his hand, and he will thoroughly purge his floor, and gather his wheat into the garner; but he will burn up the chaff with unquenchable fire." Likewise he will seperate the sheep from the goats, as he divided the light from the darkness in ages past. Do we suppose that he is come to give peace on earth? nay, but rather division. (Luke 12:51)
5) In regards to the question of exalting Jerusalem, scripture is clear that it is God's favored city throughout the Old Testament, and even in the New, Jesus himself said "O Jerusalem, Jerusalem that kills the prophets and stones them which are sent unto thee, how often would I have gathered thy children together, even as a hen gathereth her chickens under her wings, and you would not." (Matthew 23:37) Notice that he would gather them EVEN THOUGH they killed the prophets and stoned the messenger. Why would he forgive the sin and spilt blood of Jerusalem, yet not have mercy on Babylon, except that Jerusalem is favored? The true answer is my next point, which is my last.
6) Ex. 33:19 "... I will be gracious to whom I will be gracious, and will shew mercy on whom I will shew mercy." (compare to Romans 9:14-18 )
Rom. 9:18 - "Therefore hath he mercy on whom he will have mercy, and whom he will, he hardeneth."
Rom. 9:21 - "Hath not the potter power over the clay, of the same lump to make one vessel unto honour, and another unto dishonour?"
Therefore I would state that the psalmist, anointed by God's spirit (for who can pen the Word of God without his spirit), had discerned the nature of Babylon, and was aware of its fate according to God's will.
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