It was relegated finally to a particular priest with permission from the Bishop in order to keep the people from acting upon superstitious ideas. There was evidence in early Christianity that laypeople did exorcisms. In the Bible, exorcisms were done by a variety of people. One can refer to Matthew 10, Mark 9:38, Luke 11:18-19, Acts 16, 19:11-16. We can also look at Justin Martyr, Minutius Felix, Origin, and Tertullian. I believe Luther's theology would suggest that it is your baptism that makes you "eligible" to do an exorcism.
That being said, I recommend the local orthodox Lutheran pastor (no matter his past experiences) who receives good counsel from an experienced exorcist should be the one who acts as exorcist.