You can wear the mask, the cloak, the coat, the gloves, everything the doctors of that time wore, but you won't be called one for one simple reason. Not that because they would look creepy today. In those times without the hat, you won't be a doctor.
It was their ID at that time. The wide-brimmed leather hat is the thing that indicated their profession. It might sound absurd to you now, but at that time, that was the way of things. In the 17th century, every job had an official hat. The lawyers had their wigs, the clergy had their headgear, and for the doctors, it was the hat. Without one, you won't be one. Though the whole attire already gives it away.