By virtue of their office, they are in order (τάξις) different from other Christians.
Ministry in its strictest and most particularist sense has to do with the Church.
There is therefore no need to “engage in ministry” if one desires to “integrate faith and work.” The integration comes in standing for Christ in word and deed wherever one is at.
One does not need to be in “ministry” in order to integrate faith and work!
God calls various peoples to various callings in life.
We see John the Baptist for example endorsing the vocations of being a tax collector and a soldier (Lk. We note here that they were not called to join John the Baptist in his ministry, or help him out in one form or the other. In deed, believers glorify God in their deeds and works.
Rather, they were called to do justice in their respective vocations. God is the God of creation, and a carpenter putting his effort into making good wooden furniture can do so for God’s glory.