Is it permissible for a young man to expose his major 'awrah to his maternal aunt (a dermatologist) for the treatment of a skin condition, knowing that he feels uncomfortable going to a male doctor he doesn't know due to his stutter?

It is not permissible for your maternal aunt to uncover your major or minor 'awrah for treatment, except out of a dire necessity according to Sharia. Your situation is not considered a dire necessity for that. Seeking treatment from a non-Muslim male doctor is more appropriate than uncovering oneself before a Muslim female doctor. Your maternal aunt or any of your family members can inquire about a trustworthy doctor.