Would it be appropriate to ask depressed brother's professor to reach out to brother?
"I plan on contacting the prof, either via email or phone. What I ask you is for advice on how to tactfully and effectively make this ask."


If you can wait until January, I would.  Your email would seem less intrusive after the break, I think.
I suggest you make a connection with the professor, and have a rather open-ended call, rather than making a particular request.  Trust the professor's empathy and interest in your brother to (a) perhaps ask if there's anything he can do, and (b) think creatively about what he could do.

The professor is a different person from you, and his relationship with your brother is different from yours.  He may have a different way of interacting or influencing your brother, that might not be exactly what you are imagining.  But if the professor is interested in getting in touch with your brother, I imagine you'll feel relieved and grateful, and won't mind if he doesn't take a precise Jazzie3-clone approach.
I don't know what the professor's reaction would be, but if I were in his shoes, I would go into a conversation with my troubled student with, again, an open-ended approach.
Sometimes it's helpful to go with a loved one to a regular medical doctor.  See this other answer.
Sometimes it's helpful for the frustrated person (you) to talk things over with a therapist.  This can be hard to accept, but surprisingly, it can be helpful.
Possible starting point for an email:


  Dear Prof. X,
  
  I'm Zak's brother.  He was really touched when you nominated him for a
  prize, and I was impressed at how engaged he was in your class.
  
  Zak isn't doing so well this year.
  
  I can imagine that with confidentiality rules, there's not much you
  can say, but it would still be helpful for me to talk with the one professor Zak made a good connection with.  May I make an appointment to come and see you?