Separation of church and state is a theory. It implies each institution has a distinct but separate purpose. It allows believers to co-exist with non-believers in a functional government. It is the part of the Constitution that guarantees us freedom of religion.
In the end you can't take the religion out of a religious person just like you can't expect or demand a non-believer to uphold Christian values. Example, if we loaded the SCOTUS with pro-life evangelicals Roe v Wade would be overturned. The court is still functioning as the state but it's decisions will be a reflection of the people who occupy the bench.
In a pure Christian nation, Government will be a reflection of Christian values because even though state will be "separate" from the Church, there is no other input aside from followers of Christ. This is why I don't find it acceptable for Christians to hide support for worldly values behind the word "Government". A Christian should always have Christian values even if Government is totally evil.
This also explains why a Christian can vote for a candidate that is 51% good and 49% evil. You don't agree with or support the 49% evil agenda, you support them for the 51% good then pray that God will intervene on the 49% evil. If we were to use Obama as an example, you will note he didn't make the Freedom of Choice act priority like he said in his campaign. You will note his position has softened to tolerance of each other when he gave the Notre Dane address. I believe it's because we have a God that sure will answer prayers.
So yes you can separate Church from a non-believer but a true believer can't be separated from that which they believe.
Ro 8:35 Who shall separate us from the love of Christ? shall tribulation, or distress, or persecution, or famine, or nakedness, or peril, or sword?
36 As it is written, For thy sake we are killed all the day long; we are accounted as sheep for the slaughter.
37 Nay, in all these things we are more than conquerors through him that loved us.
38 For I am persuaded, that neither death, nor life, nor angels, nor principalities, nor powers, nor things present, nor things to come,
39 Nor height, nor depth, nor any other creature, shall be able to separate us from the love of God, which is in Christ Jesus our Lord.