How much influence should each have on the other?
We're the Reformers correct in appealing to their respective emporors?
Or were the Anabaptists Correct in their approach to government?
Depends on the context. While we are not
of this world, we remain in the world as agents of God's revelation and part of that revelation includes Romans 13:1
Romans 13:1
Every person is to be in subjection to the governing authorities. For there is no authority except from God, and those which exist are established by God.
That means the relationship between church and state differs from place to place and the governing authorities God established in that locale. Here, in America, we live in a pluralistic society that is governed by a representative republic where the Constitution is the preeminent rule of law (lex rex) That Constitution stipulates the (federal) government is prohibited from making any law that restricts the free express of the individual's religion. That includes the establishment of a state religion. In other words, by default, the implicit assertion is that religious diversity is protected and to be expected. Equally true, however, is the fact that America more than 80% of Americans are theists and more than 65% of those are Christians (down from 90% in the 1990s!).
None of that may exist in a monarchy (rex lex), or where those forms of democracy exist that do not limit the government's rule over religion, or a given religion is not predominant in the populace.
There exists in America an enormous unjust hypocrisy because, despite the Bill of Rights' First Amendment, the courts have also perverted some
words Thomas Jefferson wrote to decide there is a separation of Church and State, not a separation of Church
from the State.
Leaders in a New England Baptist congregation wrote to Thomas Jefferson, during his tenure as POTUS, expressing their concern that religion might be considered a right granted to the individual by the state and not a right that is inalienable (incapable of being changed by the government).
"It is not to be wondred at therefore; if those, who seek after power & gain under the pretence of goverment & Religion should reproach their fellow men—should reproach their chief Magistrate, as an enemy of religion Law & good order because he will not, dares not assume the prerogative of Jehovah and make Laws to govern the Kingdom of Christ."
President Jefferson's response was the now infamous letter to the Danbury Baptist Church in which Jefferson assured the elders the government CANNOT do such a thing simply because there exists a wall separating the government from doing so - the First Amendment of the Bill of Rights.
Unblessedly, times change and with those changes so too does the understanding and allegiance to what for one generation is an absolute. Shortly after the Civil War the US Supreme Court began to take a different view of Jefferson's now proverbial "wall" of separation, deciding religious influence on the government was to be limited, not only government legislation of religion. Furthermore, while
legislation restricting the free expression of religion is prohibited.....
influence is not. These two avenues of change gained momentum with the rise of modern liberalism, the government provision of public education, and the influx of religious diversity following two World Wars (and lesser conflicts in other countries). Some reversal of that trend was accomplished with the rise of Evangelical fundamentalism and the influence of theologians like Francis Schaeffer, but that influence was limited in scope and power and its force was not enduring. Concerns about postmodernism were replaced with post-postmodernism. We currently reside under a bane of all democratic systems; the tyranny of the one percent (Schaeffer warned the Church about this, along with many, many other philosophers over the last two centuries outside of the Church).
If, for example, the US Senate and the US House of Representatives were 100% populated by Christians (authentic converts to Christ as God views that condition) that would 1) be an unconstitutional anomaly, and 2) not likely solve much, despite the imaginations of some that is the goal. The first problem would be the failure to represent to the nation's pluralism (both religiously and politically). The second problem is demonstrated on a daily basis anytime a large group of Christians get together to decide on policy. Sample ten members of this forum and try imagining them reaching a consensus on ANY matter, not just those that are the most important to civil society.
While much more could be said about this, I'll stop for now with two last veins of commentary. God, the Creator, has created creation with an enormous amount of diversity, that diversity exists at all levels of creation, and each constituent element of that diversity is a reflection of Himself in some way or another. Much of it is adulterated by sin, but that's a sub-topic for another post. We should, therefore, expect diversity in governance, as well as diversity between one group and another, one city and another, one county and another, one state.... Of all the forms of government possible, it is the representative republic form that God established among His people (repeatedly), and neither the theocracy, the monarchy (1 Sam. 8), or the straight democracy. Despite His repeated establishment of republican (not Republican) governance, God also permitted, tolerated, and worked within all other forms throughout human history.
Lastly, the principles of scripture always more powerful and successful than the letter of scripture. Correctly understood, they never conflict with one another. Jesus' (along with the New Testament authors' shared example) repeated practice was to apply laws of God and the Law of Moses in principle, not with legalism. Legalism Kills. It kills the body politic just as much as antinomianism. What Jesus (and the other NT writers) practiced is nowadays called
theonomy but, sadly, that has become misunderstood, misused, and abused (as has every form of governance in which theonomy could be correctly applied). Most Christian criticisms of theonomy critique Reconstructionist theonomy, not theonomy in and of itself, or that not held by Reconstructionist Dominionists and that, imo, is a shame simply because New Testament Scripture repeatedly applies God's laws in principle over the letter. One notable example from scripture would be the way Jesus and the NT writers applied Deuteronomy 25:4's "
You shall not muzzle the ox while he is threshing." That Law is explicitly cited twice by Paul (1 Cor. 9:9; 1 Tim. 5:8) and implicitly cited by all four of the gospel writers and James, but not once are any of those references related to oxen or grain threshing. Taken as a whole, God's jurisprudence is restitutional and rehabilitative, not vindictive and retributional (a day has been set for the latter), but we humans screw that up chronically. Here's the fundamental rub: Any legislator, POTUS, or Justice that is elected to represent the people who does not understand this is not going to legislate, lead, or judge well. It will not matter whether they are Christian, or not.
Despite my not being a citizen of any other country, living under any other form of government, it is possible to still speak to those forms of government using what I've described above. The rock-bottom footers-to-the-foundation reality is that any country with any kind of government that is populated by God fearing, God loving people who actually live by that fear and love will be more successful than a country inhabited by godless people under godless governance. Human history is littered with the remnants and cultures who've failed in that regard. The US may one day be one of them

. Dispensational Premillennialists assure us that is the case

, but then again, they think the Church will become impotent and need rescuing

.