What is the process for re-obtaining the
medical certificate after treatment is completed?

That rather depends whether he tells the FAA or not.....

That's the key. There are 2 issues here that the FAA will react to - the
depression that is being treated and the medication that is being used.
While it may be reasonable for the FAA to specify what medications are
safe/unsafe, I don't think it is reasonable to have them wading around in
psychological conditions.

Fortunately, you do have the room to manage your health and the FAA by
maintaining separate medical relationships. But with that ability comes a
great deal of responsibility.