Public Rites of Confession and Absolution

This is a good question, and is ultimately something for your church Council to make a decision on. I'll share a few thoughts.

1) We have traditionally shied away from calling it "absolution" in our worship. We tend to use the term "assurance of pardon" along with some statement about the forgiveness of Christ offered to all those who repent of their sins and trust in them. Whoever is leading the assurance of pardon isn't declaring that people's sins are forgiven, they are sharing our understanding of Christ's forgiveness as a reassuring word.

2) Neither the assurance of pardon, nor the prayer of confession are listed in the "official acts of ministry" in the supplement to article 53 of the church order ("the administration of the sacraments, the pronouncement of blessing for the people, the laying of hands on new leaders, and the reception and formal dismissal of members"). From my understanding that means it is not a part of the service which is ordinarily limited to ordained ministers. 

3) That begin said, the Council/Consistory should still be responsible for regulating the worship service. This means what's read by the leader during the confession and assurance should fit our understanding of what we are doing at this time. It also means the person who leads this part of the service should be accepted by that same group of people. Maybe in your congregation it's only officebearers and ministers who conduct any prayers during the service. In that case, for consistency it would be good if an officebearer or minister led the prayer of confession and then shared the assurance of pardon. Maybe in your congregation a variety of lay-people offer prayers during the service. In that case there's no reason why a lay-person can't also lead this part of the service.