Actually, we do have a very good idea of when the State is acting as an instrument of God. It does so when it punishes the evildoer.
St. Paul himself said so.
Romans 13:3-4
Aquinas used that very passage when he noted that it was permissable to kill a person to safeguard the community
-ST IIa-IIae, q. 64, a. 2.
Which the Angelic Doctor followed with:
-ST IIa-IIae, q. 64, a. 3.
If I recall correctly, St. Paul taught that all Christians should be celibate, turning only to marriage if they were too weak.
St. Augustine wrote that it was the duty of every bishop to plead for mercy for each person condemned to death.
And St. Thomas Aquinas agreed with St. Jerome, Pope Innocent III, and St. Augustine that abortion was not murder until the fetus had assumed a human form.
This is why we are not Protestants. These are saints, tremendous examples of the faith. But they were still men. We believe that Jesus was/is God, so what did he say?
Unfortunately, on the death penalty, not much. He did not object to civil authority, but he also advised that he without sin should cast the first stone. He also openly questioned applying an eye for an eye.
So, how do we decide what to follow and what to disregard? As a Catholic, I have it easy. I follow the Holy Catholic and apostolic Church:
“Among the principal duties of bishops the preaching of the Gospel occupies an eminent place. For bishops are preachers of the faith, who lead new disciples to Christ, and they are authentic teachers, that is, teachers endowed with the authority of Christ, who preach to the people committed to them the faith they must believe and put into practice, and by the light of the Holy Spirit illustrate that faith. They bring forth from the treasury of Revelation new things and old, making it bear fruit and vigilantly warding off any errors that threaten their flock. Bishops, teaching in communion with the Roman Pontiff, are to be respected by all as witnesses to divine and Catholic truth. In matters of faith and morals, the bishops speak in the name of Christ and the faithful are to accept their teaching and adhere to it with a religious assent. This religious submission of mind and will must be shown in a special way to the authentic magisterium of the Roman Pontiff, even when he is not speaking ex cathedra; that is, it must be shown in such a way that his supreme magisterium is acknowledged with reverence, the judgments made by him are sincerely adhered to, according to his manifest mind and will. His mind and will in the matter may be known either from the character of the documents, from his frequent repetition of the same doctrine, or from his manner of speaking.” - LUMEN GENTIUM (Dogmatic Constitution of the Church)
When the Vicar of Christ comes to the US and calls for an end to the death penalty, and the princes of the Church include this call in the local Catechism, I listen.
If others are impeded from obeying by the absolute certainty of their moral conscience, it is not my place to judge them. But, as the Catechism warns, they should be mindful that their own certainty may be incorrect. In Catholicism, the benefit of the doubt on proper application of the faith always belongs with the Church.