What safeguards are in place to prevent abuse of power? In the U.S., what safeguards are in place to prevent a mentally ill, or just spiteful or incompetent, psychiatrist / psychologist from abusing their position of power?
Assuming the person is already practicing.

For example, are there any diagnoses that require an independent second opinion because of the serious consequences of acting on that diagnosis?

Obviously, this is especially important since- given the near complete power someone in this field could have over another person's life trajectory- the field will attract abusive people.
As @ArnonWeinberg mentioned, clinical supervision, detailed in my answer to the previous question will prevent abuse of position.

Clinical supervisors have a duty of care, not only to the supervisee (the therapist), but also the supervisee's clients, and if the supervisor feels that abuse of power is occurring they must refer it to the governing bodies to remove their licence to practice.

If they feel there are any mental health issues preventing the supervisee from acting effectively, then the supervisor will refer the supervisee for therapy. If necessary, they may also take further action to prevent the therapist from causing harm to their clients whilst undergoing therapy. I have added this point to my previous answer mentioned here.