Bear in mind that Judaism in antiquity was a diverse religion and different Jews had different understandings of what "the Law" was and what it meant. Also bear in mind that no-one can agree on what "the Pauline" view of "the Law" is either. And that if you read the Hebrew Bible from a Christian point of view you're almost certainly going to misunderstand it. For example:
The Jews were well aware of this, as the psalm in question indicates. I think you're assuming too much of a distinction between Judaism and Christianity. What if what you think of as the Christian attitude towards the Law was actually pretty similar to how many Jews already thought of the Law?
Which part of Judaism, and which part of Christianity? I doubt there is much distinction between them when both thought they were carrying out God’s governance on earth. The underlining theory of sin, how humans react to their own nature, how one handles confession, and how atonement was administered, never changed since Adam until now. The Law of Moses never addressed that. The Law was a set of rules to provide an economy related to this condition. It is not even certain that the Jews gave any constant attention to the administration of the Law, other than the faithfullness of the scribes to keeping the record in tact. And even that has been attacked as never happening, because at times it was said to be lost, because the Jews were constantly ignoring it and doing their own thing.
That David figured it out was not happenstance. He was considered a man after God's own heart, despite killing humans and committing horrible acts. Neither did he or his family "get away" from the consequences. That he was even king was the result of the Jews rejecting the Law, and demanding a kingdom as the form of governance. There was even supposed to be a separation between the kingship and the duties of the priesthood. And there was never a sacrifice for murder. The atonement was death. David could not offer one because he was king, there was no sacrificial atonement for murder, and God himself told him he could not build a temple, much less offer up sacrifices in it. Which term of condition was he even referring too? Not to mention we have the evidence that the only sacrifice for the human nature of sin was God himself in the form of the first begotten Son. I am not even convinced that what happened after the resurrection could even be considered a sect of Judaism. For one, no group had the full knowledge of what God's plan even was. There was too much human ideology mixed into the core of what David was posting as his thoughts. What happened in the next 200 + years after the advent, seemed to indicate a mix of kingdom mentality, Judaism, and Christian authority and due to the democratic nature of that authority evolved into an ideology all of it's own, thus creating a governing religious body. All involved were trying to reason out what was happening and come to an acceptable conclusion. Even though that was not even the intent that Jesus himself gave. The condition was not settled. God's redemptive work was. So keeping the Law, doing righteous deeds, etc. worked out one's salvation, but such actions did not change the condition nor satisfy the atonement demanded. The theme of this chapter was rightly acknowledged as a model confession. As far as I can find, this chapter was never used as proof of original sin. And David still had the conviction that offering up sacrifices was the means to work out his salvation along with keeping the perfect Law of God. Salvation has always been given to us after confession, and never something we generate on our own.
Even pagans are forced to deal with the human condition, and come up with inventive ways to deal with all the consequences it promulgates.