I wouldn't use either, with a few exceptions.
You don't know what's in the water (often recycled) at the auto wash, nor do you know what kind of "soap" is being used.
As for the bucket, the only reason to wash a car with Dawn is to take off all the wax, and unless you are going to do that one time to apply a polymer finish (e.g.: Zaino,which will not bond to the paint through a wax), using Dawn and a bucket is a pretty harsh way to treat any car.
If you want to treat the car "right from the start," you need to spend some time here reading up on how best to wash a car without damaging the paint, and how best to protect the paint by using either wax or a polymer.
Get the right protectant, the right soap, the right cleaning implements and the right drying towels (usually microfibre these days), and leave the Dawn up in the kitchen.
