It turned out that the door opener was under powered for such a big door so it moved up much slower than the one on the car side which I am familiar with.
Am I understanding what you are saying correctly ?
… you drove forward as the overhead door was still in motion, lifting up ?
If I am understanding you correctly, then the damage occurred when you moved the MH forward ? Misjudging the lifting speed of the doors motor .
If damage occurred due to the movement of your vehicle , then it would technically be an auto accident and that is why you decided to work this out on your own with the car wash owner. Even though it's on private property BUT has public access, so the rules of the road apply.