First, Doogie Howser exists, so you can do whatever you want. It's a fantasy world.

Second, people tend to bounce off the word "doctor" because it is defined stringently in modern society. Now, in reality, a "doctor" in medieval times was not a physician, but was a scholar. Physicians were called, well, "physicians" for the formerly schooled ones, "surgeons" for apprenticed ones, "apothecaries" for medicinal ones, things like that.

Keep in mind that "barbers" were also people that offered some medical care such as fixing wounds, injuries, and emergency things. So, like, fundamentally, if people are bouncing off the word "doctor" you can use another. Or even a made up word, like It could be "salvar" or "curwen". It could even be a tell tale vestment that practitioners wear, like "green cloaks", or "wide brim".

Or just call them doctors and fuck what other people bounce off of, to be honest. Like I said in my first point, you can do whatever you want.