I think it is quite clear Paul is indeed talking about the Law of Moses:
What shall we say then? Is the Law sin? May it never be! On the contrary, I would not have come to know sin except through the Law; for I would not have known about coveting if the Law had not said, "You shall not covet."8But sin, taking opportunity through the commandment, ...
It is the Torah - the Law of Moses - that contains the commandment "You shall not covet". Paul's argument then proceeds seamlessly to argue that the Law of Moses, strangely, leads to sin and death.
You are correct that Romans 7:7-8 is referring to the Law of Moses, but the phrase “the Law of Sin” does not. It is “another law” (Romans 7:23):
Rom 7:22 For I delight in the law of God after the inward man:
Rom 7:23 But I see another law in my members, warring against the law of my mind, and bringing me into captivity to the law of sin which is in my members.
Rom 7:24 O wretched man that I am! who shall deliver me from the body of this death?
Rom 7:25 I thank God through Yeshua Messiah our Lord. So then with the mind I myself serve the law of God; but with the flesh the law of sin.
Paul’s desire was to obey the Law of God which, in reality, is the Law of Moses/Torah. Yet, his flesh in his unconverted state (which is what he is portraying in his example) cannot obey it. His weak flesh cannot subject itself to Torah.
Rom 8:6 For to be carnally minded is death; but to be spiritually minded is life and peace.
Rom 8:7 Because the carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be.
Rom 8:8 So then they that are in the flesh cannot please God.
Rom 8:9 But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Messiah, he is none of his.
Once the carnal man who cannot be subject to the Law receives the indwelling Holy Spirit, he can obey Torah. The Spirit of YHWH has power over the flesh so that the spiritual man living under the New Covenant will not only fulfill the righteousness requirements of Torah (Romans 8:4), but the Holy Spirit will cause him to obey it.
Eze 36:26 A new heart also will I give you, and a new spirit will I put within you: and I will take away the stony heart out of your flesh, and I will give you an heart of flesh.
Eze 36:27 And I will put my spirit within you, and cause you to walk in my statutes, and ye shall keep my judgments, and do them.