The context of the Psalm does not need to be Babylonian. Commentators ascribe a wide range of possible moments for its composition, from the days when the First Temple still stood, to the Exile, to the Maccabean revolt in the 2nd century when the Jews were temporarily defeated by the forces of Antiochus IV (as per John Calvin and others).
The most likely pre-exilic setting for the Psalm would be during the reign of Hezekiah, one of the "good kings." He barely staved off conquest by Sennacherib of Assyria, whose armies successfully attacked every major city of Judah except Jerusalem.
In the fourteenth year of King Hezekiah’s reign, Sennacherib king of
Assyria attacked all the fortified cities of Judah and captured them.
So Hezekiah king of Judah sent this message to the king of Assyria at
Lachish: “I have done wrong. Withdraw from me, and I will pay whatever
you demand of me.” The king of Assyria exacted from Hezekiah king of
Judah three hundred talents of silver and thirty talents of gold.
So Hezekiah gave him all the silver that was found in the temple of
the Lord and in the treasuries of the royal palace. (2 Kings 19:13-15)
This was a truly desperate situation. Even though Hezekiah had done some wrong, the Covenant had not been broken. Earlier, Assyria had conquered the northern tribes and forced many of them into exile. According to his annals, Sennacherib took 46 fortified cities of Judah and small cities beyond number. He also reports "leading off" an enormous number of prisoners:
As for Hezekiah, the Judean, he did not submit to my yoke. I laid
siege to 46 of his fortified cities, walled forts and to the countless
small villages in their vicinity …. I led off 200,150 people, young
and old, male and female, horse, mules, donkeys, camels, big and small
cattle beyond counting, and counted them as booty.
Thus, although the phrase "you scattered us among the nations" was applied to other situations where the Covenant had indeed be broken, it might have been used here to describe 200,000 Judeans who were "led off" by Sennacherib after the northern tribes had already been scattered. If so, the psalmist was right to say that this had happened even though the Covenant was still intact.
But if the context is indeed Babylonian, then the OP question is apt. The psalms were not written by just two or three people, and it logically possible that a few of the psalms do not reflect the traditional idea that the Exile was a just punishment for the Judeans breaking their Covenant with God. After all, the Bible includes the Book of Job. In that scripture the Lord declares that Job "is blameless and upright, a man who fears God and shuns evil." Yet God removes his protection from Job, and Job suffers terribly as a result. Job dares to ask:
How many wrongs and sins have I committed?
Show me my offense and my sin. Why do You hide your face
and consider me your enemy? Will You torment a windblown leaf? (13:23-25)
If the blameless Job could utter those words, perhaps the author of Psalm 44 did likewise.
In the end we are left with two possibilities: either Psalm 44 actually refers to a situation where large numbers of Judeans where captured and led off, or it is a Job-like questioning of what the author thinks is an unjust punishment. For those who find the latter option unacceptable, the explanation that it refers to the time of Hezekiah is a sensible alternative.
APPENDIX
From Calvin's Commentaries Vol. 9:
It is uncertain who was the author of this psalm; but it is clearly
manifest that it was composed rather by any other person than by
David. The complaints and lamentations which it contains may be
appropriately referred to that miserable and calamitous period in
which the outrageous tyranny of Antiochus destroyed and wasted every
thing. [prior to the Maccabean Revolt] Some, indeed, may be disposed to apply it more generally;
for after the return of the Jews from the captivity of Babylon, they
were scarcely ever free from severe afflictions. Such a view,
doubtless, would not be applicable to the time of David, under whose
reign the Church [meaning the people of God] enjoyed prosperity. It may be, too, that during the
time of their captivity in Babylon, some one of the prophets composed
this complaint in name of all the people.
