As a doctor in practice for several years I don't know why
"flashing" and M.D. badge would or should help you.
Should nurses flash "RN" or "LPN" or "BSN" and expect special treatment?

Would you tell them that you're "in training" as a resident? Because that would be the truth.
I frankly would be embarassed to even ask - why should we receive special treatment, especially when most of us can afford to buy the ticket outright?

If they need help, they'll ask. I've flown hundreds of thousands of miles and have never been onboard when they've even asked for a doctor.

You're a resident, and I have some advice for you - don't flash your professional status to the public.
It's a profession, not something to "flash", and you'll appear very arrogant.
The truth is, nobody really cares that you're (we're) doctors, unless they're patients. Trust me on this.
Don't dishonor the profession. Television is already doing a good job at that.

I think you'll be able to afford the business class/first class ticket in a few years - be patient. It's obvious you have experienced delayed gratification, or you wouldn't be where you are today.

Good luck in your studies.
Maybe you should consider investment banking? It's never too late.
The world needs more investment bankers and they make LOTS of money, right? That comment of yours was shameful!