In English the title "Doctor" is the same whether or not the person is male/female, but in German there are two different words for this.
Not sure about that plot device, but it should work the same in German. Yes, there is Doktor and Doktorin in principle, but Doktorin is rarely used.
In practice, when addressing either male or female doctor by name and with title, you would in both cases say, for example, Doktor Müller. If you address a Doctor only by title, then you would say Herr Doktor for men and Frau Doktor for women.
If you talk about a doctor in his/her absence, then you have the choice between all three forms: Doktor Müller, or der Doktor/die Doktorin or "Herr/Frau Doktor".
Also, now the word Doktor, having it written so often in so few sentences, has lost all meaning and just sounds really weird...