Let's consider this perspective.
Romans 1: 7 “To all in Rome who are loved by God and called to be his holy people:”
By writing this verse, Paul broke Roman laws. At that time, it was against Roman law to preach different religions to Roman citizens.
Romans ruling over Judea seemed to think it was ok for Jews to "be alive" and have synagogues. You see those synagogues all over the Roman empire at the time of Acts.
In Acts 16:16-34 we find that Paul was accused of “advocating customs that are unlawful for…Romans to adopt or observe.”
Acts 16 - out-in-the-open monotheistic Bible based worship
12 and from there to Philippi, which is a leading city of the district of Macedonia, a Roman colony; and we were staying in this city for some days. 13 And on the Sabbath day we went outside the gate to a riverside, where we were supposing that there would be a place of prayer; and we sat down and began speaking to the women who had assembled.
Even demons publicizing that these were mono-theistic servants of God- did not get the Romans to oppose them in Acts 16
16 It happened that as we were going to the place of prayer, a slave-girl having a spirit of divination met us, who was bringing her masters much profit by fortune-telling. 17 Following after Paul and us, she kept crying out, saying, “These men are bond-servants of the Most High God, who are proclaiming to you the way of salvation.” 18 She continued doing this for many days. But Paul was greatly annoyed
Notice it is not Romans that are "greatly annoyed" at that - but Paul.
NOW those who object are the pagans who were in the business of slave trade selling the services of the demon who possessed the girl.
They were the ones coming up with fake accusations.
19 But when her masters saw that their hope of profit was gone, they seized Paul and Silas and dragged them into the market place before the authorities, 20 and when they had brought them to the chief magistrates, they said, “These men are throwing our city into confusion, being Jews, 21 and are proclaiming customs which it is not lawful for us to accept or to observe, being Romans.”
Notice they do not complain "our hope of profit is gone - they cast out our demon"
And end the end the authorities begged Paul to leave peacefully --
35 Now when day came, the chief magistrates sent their policemen, saying, “Release those men.” 36 And the jailer reported these words to Paul, saying, “The chief magistrates have sent to release you. Therefore come out now and go in peace.” 37 But Paul said to them, “They have beaten us in public without trial, men who are Romans, and have thrown us into prison; and now are they sending us away secretly? No indeed! But let them come themselves and bring us out.” 38 The policemen reported these words to the chief magistrates. They were afraid when they heard that they were Romans, 39 and they came and appealed to them, and when they had brought them out, they kept begging them to leave the city. 40 They went out of the prison and entered the house of Lydia, and when they saw the brethren, they encouraged them and departed.
So I don't see it as making your case - but even so Rom 13 does not say "only if Rome is perfect, only then obey". Rather in all its corruption Paul still said to obey civil laws so far as conscience informed by scripture allows.
Romans: 13: 1 “Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God.”
In this verse, Paul challenges Caesar's authority.
I don't see any challenge to Caesar.
When Christ was put on trial the POV of Pilate was not "you are a Jew, a monotheist - I condemn you" -- rather it was "I find no fault in Him"