The Book of Concord is the seminal reference for Lutheran theology, and is a collection of statements from the period. Some Lutherans ascribe to all of the BOC, others argue about applicability of various documents beyond the Augsburg Confession. Here one relevant text from the Augsburg Confession,
Article XVI: Of Civil Affairs.
1] Of Civil Affairs they teach that lawful civil ordinances are good works of God, and that 2] it is right for Christians to bear civil office, to sit as judges, to judge matters by the Imperial and other existing laws, to award just punishments, to engage in just wars, to serve as soldiers, to make legal contracts, to hold property, to make oath when required by the magistrates, to marry a wife, to be given in marriage.
3] They condemn the Anabaptists who forbid these civil offices to Christians.
4] They condemn also those who do not place evangelical perfection in the fear of God and in faith, but in forsaking civil offices, for 5] the Gospel teaches an eternal righteousness of the heart. Meanwhile, it does not destroy the State or the family, but very much requires that they be preserved as ordinances of God, and that charity be practiced in such 6] ordinances. Therefore, Christians are necessarily bound to obey their own magistrates 7] and laws save only when commanded to sin; for then they ought to obey God rather than men. Acts 5:29.
Here we have an explicit statement that individual Christians may hold state office, in the Augsburg Confession. The BOC continues with Defenses and Solid Declarations, which essentially repeat the explanation in the AC.
The Sixteenth Article [of the Augsburg Confession] the adversaries receive without any exception, in which we have confessed that it is lawful for the Christian to bear civil office, sit in judgment, determine matters by the imperial laws, and other laws in present force, appoint just punishments, engage in just wars, act as a soldier, make legal contracts, hold property, take an oath, when magistrates require it, contract marriage; finally, that legitimate civil ordinances are good creatures of God and divine ordinances, which a Christian can use with safety. 54] This entire topic concerning the distinction between the kingdom of Christ and a political kingdom has been explained to advantage [to the remarkably great consolation of many consciences] in the literature of our writers, [namely] that the kingdom of Christ is spiritual [inasmuch as Christ governs by the Word and by preaching], to wit, beginning in the heart the knowledge of God, the fear of God and faith, eternal righteousness, and eternal life; meanwhile it permits us outwardly to use legitimate political ordinances of every nation in which we live, just as it permits us to use medicine or 55] the art of building, or food, drink, air.
This statement here makes reference to Luther's earlier writing on Two Kingdoms with "in the literature of our writers."
Twice the BOC explicitly states that "a Christian" may serve in civil office. Any speculation that there could be some distinction in Lutheranism between what a pastor may do and what "a Christian" may do can easily be disproven by reading the appropriate sections.
To explain it further, consider someone with dual vocations of pastor and sheriff. I've seen Two Kingdoms described as "hat you're wearing." Such a dual-vocation person would proclaim forgiveness of sins in church, and arrest bad guys in the saloon. It is about separation of jobs. Our hypothetical person shouldn't arrest people for failing to attend church, nor should he apply forgiveness of sins during the arrest of a bad guy.