Romans 13: 1 1 1 Let every person be subordinate to the higher authorities, for there is no authority except from God, and those that exist have been established by God. 2 Therefore, whoever resists authority opposes what God has appointed, and those who oppose it will bring judgment upon themselves. 3 For rulers are not a cause of fear to good conduct, but to evil. Do you wish to have no fear of authority? Then do what is good and you will receive approval from it, 4 for it is a servant of God for your good. But if you do evil, be afraid, for it does not bear the sword without purpose; it is the servant of God to inflict wrath on the evildoer.
5 Therefore, it is necessary to be subject not only because of the wrath but also because of conscience. 6 This is why you also pay taxes, for the authorities are ministers of God, devoting themselves to this very thing. 7 Pay to all their dues, taxes to whom taxes are due, toll to whom toll is due, respect to whom respect is due, honor to whom honor is due .
Paul was pretty clear.
Yes he was.
Those secular authorities have an area where they are permitted to exercise power, As you said “Caesar’s things”, but it was constrained by God”s law so it was only a relative power and authority hence Paul uses the word “tetagmenai” which means to set in a certain order, to assign a place.
Peter said Christians were to “obey God as ruler rather than men” when that God permitted authority overstepped it’s bounds.(Acts:29)
Paul called the secular authorities God’s “servant, diakonos, ministers” to the extent that it did enforce good conduct and order. Paul thus could call upon the Roman government, no matter how evil it was in other ways, for protection.
The approval seems to have been from the authorities for the good Christian conduct not the other direction.
“Then do what is good and you will receive approval from it,”