Not really, no. The entire New Testament takes place during a time Christianity was a weird sect of Judaism, and at best ignored by Rome (persecution was by Jewish authorities). Not sure what the 'foundational act' of Christianity would be, maybe Jesus' death and resurrection, maybe something in the Book of Acts (
badum-tsh, but like, Pentecost), but it wasn't caring about Roman governance.
Christ didn't critique Rome. He didn't really have words for Rome at all, beyond "Give to Caesar what is Caesar's". The threat he embodied to 'the system' was specifically the Judaic priesthood who didn't view him as the Messiah. Well, arguably he embodied a broader threat to Rome, in that as you say, living in ways counter to Rome (and 'the world' at large) it challenged institutional assumptions and gave way to reform, but that wasn't a threat Rome really recognized, and, key to your point I think, was not trying to affect Roman law, nor lead to it even amongst those who were citizens of Rome.
Other than the aforementioned "give to Caesar what is Caesar's", which certainly doesn't seem to challenge Rome, or indicate any political stance beyond "pay your taxes", can you come up with any time Jesus even -acknowledged- Rome? He didn't even disobey Roman law; both Pilate and Herod Antipas acknowledged they could find "no guilt in him". It was a Jewish affair.
Even Paul didn't seem particular interested in Rome. He was a citizen of Rome, afforded rights, which he availed himself of when it came to who could judge him, but which he never talks about using to vote (despite as a citizen of Rome having that right, at least for local leaders). He doesn't instruct citizens to vote, let alone how they should vote, nor does he instruct them to be active in civics. He does give instruction for how to engage with civic leaders - pray for them, and obey the law.
That hardly paints a picture of getting religion into politics.
More to the point though, what I was referencing was specifically Jesus' and Paul's statements regarding the law. Namely, that all the law could do is condemn. It's part of a broader understanding that one aspect of the Old Testament is that it serves as a demonstration that even God's chosen people are unable to live up to the law; people can't be good enough. Everyone who lived under the law was condemned by the law. That trying to legislate morality just serves to punish people for being immoral. That something different is needed to actually make people righteous (Christ, sanctification through grace, etc etc). It is ironic, then, how many Christians end up deciding that what this culture really needs is more law, morality via legislation, rather than trying to affect the culture by, well, living counter culturally, loving those who persecute you, etc.
If the Christian voting bloc collectively shrugged its shoulders at abortion as a law, while at the same time showing up at abortion centers to love and support those who were there, ensuring their needs as potential mothers, as well as any needs from having an abortion (which is oftentimes still traumatic) were met, voting not to punish and condemn ("you got raped at 14? Too bad; you get to be a single teenage mother now. Have a good life") but instead to enable (mandatory paid maternity leave, welfare programs to ensure children would be taken care of, healthcare coverage, etc, so that having a child wasn't an economic burden), it would likely lead to a much better reception to their message than the current most highly visible tact of political partisanship.