It is an abomination to kings to do evil,
for the throne is established by righteousness.
I have never had time to prove all the ways this is true, but Romans 12 is stuffed with Proverbs. The point where he is acknowledge to quote Proverbs is almost misleading because almost everything else he says in chapter 12 is also from Solomon.
And if you know Proverbs, then the transition to Romans 13 makes complete sense. Paul’s perspective ™ on kings is Solomon’s view of kings. Objectively a king must practice righteousness to prosper on the throne. Evil kings there are many, but they always undermine their own dynasties.
And your job is not to talk up rebellion, but to appeal to the king’s only real source of security: you are to treat him as a judge who must do right.
Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God. Therefore whoever resists the authorities resists what God has appointed, and those who resist will incur judgment. For rulers are not a terror to good conduct, but to bad. Would you have no fear of the one who is in authority? Then do what is good, and you will receive his approval, for he is God’s servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God’s wrath on the wrongdoer.
One thing that might help here is to realize that Solomon’s perspective is multi-generational. The righteous prosper through their children and grandchildren while the wicked are removed from power and wealth in three to four generations. Perhaps I can show that in a another post some time.
(The ™ thing is kind of an inside joke. If it doesn’t mean anything to you, don’t worry about it.)