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?
One thing to be aware of is that in the US there is a law (FERPA) that limits what professors can discuss with others about students and their academic performance. The professor might not be in a position to talk about this student at all (even stating that a person was a student at the university could be a FERPA violation.)
My suggestion is to do what you think can help your brother and not worry too much about what is “appropriate”. With that said, I think you are putting too much hope in the professor’s ability to influence your brother. Depression is a serious illness that among its other effects can greatly distort people’s view of reality, to an extent that often cannot be countered using logic and persuasion (but may be addressed using medication and other medical care).
So go ahead and try if you must, but keep in mind that the professor is quite likely to refuse to intervene, and that even if he agrees, the chances that it would make any difference are not great.
Finally, I think the idea to ask the professor to suggest to your brother to go to therapy is a very bad idea. The professor cannot possibly know your brother well enough for such a recommendation to carry any authority (you said yourself that “he has no reason to know just how dire my brother’s situation is”), so making such a suggestion would only undermine his credibility and make it less likely that your brother would listen to any of his other advice.
Good luck. And just to be clear, everything I say here is my opinion only and should not be used as a substitute for professional medical advice, which I think is what your brother really needs.