Did you have a checklist? What was on it? How did your general contractor behave towards you regarding permits? All this could make for a good comparison and be helpful for the OP, right?!
We needed, besides the building permit or the "certificate" thereof, only the permit for the crane setup area. The crane company took care of this permit, which we authorized through the home construction company. On the page https://www.hausbau-forum.de/threads/Town & Country-moechte-baukran-und-transportkosten-berechnen.42105/page-9#post-575748 post #65 there is something about it. (Update: No credit was issued.)