The lounge complex has central shower facilities, and they can be quite busy at that time of morning. There were also a couple out of action when i was there a week ago.

Call You First and see if you can get a cabana in the Concorde Room. Alternatively, if you can clear immigration easily you may want to use the Arrivals lounge and the F breakfast area attached to it. There must be close to 50 shower rooms in there.