In Romans 8:4-7, those who walk in the Spirit are contrasted with those who have minds set on the flesh who are enemies of God who refuse to submit to His law, and in Galatians 5:19-23, everything listed as works of the flesh that are against the Spirit are also against God's law, while all of the fruits of the Spirit are in accordance with it. Christ is God's word made flesh, so us embodying God's word through following his example is the way to turn to him. In Jeremiah 31:33, the New Covenant involves God putting the Mosaic Law in our minds and writing it on our hearts, and in Ezekiel 36:26-27, the New Covenant involves God removing our hearts of stone, giving us hearts of flesh, and sending His Spirit to lead us to obey the Mosaic Law.
In Romans 7:22-23, Paul said that he delighted in obeying the Mosaic Law, but contrasted that with the law of sin that held him captive, so verses that are interpreted as referring to the Mosaic Law should make sense for them to be referring to something that Paul delighted in doing. For example, it would be absurd to interpret those verses as referring to the Mosaic Law as if Paul delighted in stirring up sinful passions in order to bear fruit unto death or as if he delighted in being held captive to sin, but rather it is the law of sin that he described as holding him captive.
In Romans 7:7, the Law of God is not sinful, but is how we know what sin is, however, a law that stirs up sinful passions in order to bear fruit unto death is a law that is sinful, so Romans 7:5 and 7:7 can't both be referring to the same law. When the Law of God reveals our sin, then that leads us to repent and causes sin to decrease, but the law of sin is sinful and causes sin to increase. So verses that refer to a law that is sinful, that causes sin to increase, or that hinders us from obeying the Law of God should be interpreted as referring to the law of sin, such as Romans 5:20, Romans 6:14, Romans 7:4-6, Galatians 2:19, Galatians 5:16-18, and 1 Corinthians 15:56.
In 1 Peter 1:16, is making a quote from Leviticus, so what he was quoting from is directly relevant to the point that he was making and should not be ignored. The Law of Moses was given by God and the Spirit is God, so it is the Law of the Spirit, which is why all of the fruits of the Spirit are in accordance with it, so maturing in all of the fruits of the Spirit does not refer to something other than what Peter was quoting from.