That slogan that Paul was a dual citizen of Rome and Heaven is used without analysis to wiggle out of Jesus clear teaching that you cannot serve to masters.
Paul was clear that he was a citizen of heaven and that is how he lived.
For our citizenship is in heaven, from which we also eagerly wait for the Saviour, the Lord Jesus Christ (Phil 3:20).He explained his attitude to political and religious citizenship in Phil 3:5.6. Paul considered these all crap.
Paul was recorded once as using his Roman citizenship, but that does not make it right. He was already a prisoner, and pulled his Roman citizen card to avoid another beating (Acts 22:22-30). I am not sure if the was right in doing this, as in the past he seems to have chosen to take the beating.
I have worked much harder, been in prison more frequently, been flogged more severely, and been exposed to death again and again. Five times I received from the Jews the forty lashes minus one. Three times I was beaten with rods, once I was pelted with stones (2 Cor 11:23-25).Maybe in Jerusalem he could not just face another beating. I can understand that and would probably have done the same, but that does not make it right. The problem for Paul was that it left him under the control of Rome. By appealing to Caesar, he go taken to Rome, but he lost his freedom to operate as an apostle. If he had taken the beating and been rescued by the Spirit, he would have been free to travel to Rome as an active apostle. He might have achieved more for the Lord.
Maybe God needed to put Paul into prison, to prevent the church from turning him into a Pope. Perhaps God took him out of play, so he would write more letters and open the way for others to step up.
Whatever, Paul’s decision in a moment of weakness, should not be used as a basis for a political theory.