Lutheran hour ministries has recently put a series of videos on the distinction between church and state in America. I would advise watching them if you haven't. It is particularly helpful in that the documentary seeks to approach the subject in a balanced way, affirming both the distinction between the two kingdoms, and the necessary interaction between them.
I think this is particularly helpful when looking at some of the more extreme views of supposed two kingdom theology in Reformed churches. Look for example at the following quote from Dr. Joel Beirmann in this documentary regarding the Christian necessity for cultural and political engagement,
"The error of assuming that I don't have a responsibility for this realm is an error. You could even say a heresy. Because if you get the distinction right between the two realms its easy to see that yes I have a responsibility on the left hand and and there's no way I can be quiet; I need to be active."
It becomes clear that Lutheran two kingdom theology is not the same as the American distinction between church and state; I fear that some have conflate the two concepts.