“Maestro,” of course.
What’s the “dilemma” here?
What do you mean “supposed to”?
Your question seems out of left field in the context of this thread.
I once had a colleague who had obtained an Ed.D. degree (in education administration) late in his life. He always used the title in correspondence, despite the fact that the degree had nothing to do with his work (he installed and maintained IT equipment, and never did any work remotely related to education). He went as far as to use the title when making airline reservations, claiming that the airlines liked having physicians on flights in case passengers needed medical attention. So making them think he was one provided protection against getting bumped from overbooked flights. IMO the degree was basically a vanity project for him, and that he chose a doctoral program that would be easy to complete.
Someone upthread mentioned chiropractors: bodybuilding champion Franco Columbo became a practicing chiropractor after he retired from bodybuilding. He always seemed to insist on being addressed as “Dr. Columbo” in interviews and the like. Later, he got busted for claiming on his letterhead that he held a Ph.D. in nutrition.
People who insist on always being addressed by a title remind me of used car salesmen who call themselves “Honest John”. If you’re actually honest (or learned) people will figure it out.
He was wondering whether he should introduce his friend as “Doctor So-and-So”. I find it vaguely analogous to whether he should, or would, introduce a hypothetical friend of his, an admiral, as “Admiral So-and-So”. Personally, I would introduce him by first name only, and leave it up to my friend to mention his profession/title, or for that matter not mention it, himself in the ensuing chit-chat.
If you’re his wingman you introduce him as Admiral So-and-so, the highest ranking Navy Seal, and hero of the Battle of Macho Grande.
If any of my friends don’t have “game” enough that they need my help as a wingman…then they’re SOL.
Admirals usually don’t need a wingman. Neither do medical doctors. PhDs often need help, otherwise they’ll end up having children with other PhDs, and I can attest that leads to serious brain function abnormalities.
But so can a car mechanic or plumber, but they don’t get special titles.
You’ve presumably heard of Dr. Martin Luther King, Jr? He was not a medical practitioner, who are relatively Johnny Come Latelys to the title. So PhD’s being called Dr. is appropriate in my opinion - or filling out forms as Dr. Firstname Lastname. Though insisting on being called Dr. in social settings is annoying and boorish.
I’m involved in a conference with plenty of PhDs, both in industry and academia, and no one uses “Doctor” professionally, even in introductions to talks.
The only time my title was ever used at work was decades ago when HR people sent instructions to interview candidates, including that “Doctor Voyager” would meet them for breakfast at the start of the interview day. They were mostly grad students about to get a PhD, so maybe HR thought that would set their expectations properly.
Now if a physician ever got snooty with me, emphasizing that they are a doctor and I’m not, I might tell him or her “That’s Doctor Voyager, bozo,” but so far it has never come up.
I suspect he used it (or it was used for him) to help the bigots remember that he was literate. To put it mildly. If anyone testifies to Congress, for instance, I’d expect them to use their title to stress their qualifications.
I’ve heard plenty of Pastors who have gotten Doctorates called Doctor, so I don’t think that’s necessarily it. Though some may have used it for Civil Rights purposes.
And people still do refer to him as Dr. King.
The wife has a PhD in a nonmedical field. She generally does not insist on people calling her Dr. but rather opts for Mrs. in social settings.
We had an idiot at our agency that received an honorary doctored from some rink dink church. He didn’t even have a college degree. After that, he wanted to called Doctor. No one who knew him would call him doctor, but people who didn’t would call him doctor because they didn’t know any better. He was fired about a year later for total incompetence.
We had a change to our state parking regulations because a highly educated man insisted on parking in the ‘Reserved for Doctors’ area outside of a hospital. He insisted that he was a Doctor because he had a PhD.
They changed it to ‘No parking without a permit’ and the state parking regulations were changed. Now the only places that say ‘Reserved for Doctors’ are in privately owned parking lots, where they can tow you if they feel like it.
Maybe this so-called idiot wanted to be fired!
Do any recipients of honorary degrees (from prestigious institutions) refer to themselves (or have others refer to them) as “Doctor”?
And here is the recipient of several honorary doctorates - the President of the USA - Dr. Donald Trump! 
Just to add, my understanding is that academic, nonmedical doctors is the original sense of the title, way back in the Middle Ages. Or is that not correct?
A “doctor” is a teacher or instructor. Cf. “medicus”.
When I moved to Texas, I met a retiree with common interests, we went to maybe a hundred ball games together, and birded together on a monthly survey, for 7 or 8 years, before I knew he had a PhD, has a long article in Wikipedia, and has a scientific phenomenon named after him. Never heard anybody call him Doctor…