Which of the following two expressions is better?
- A doctor by training as he is, he has proved himself to be a successful writer.
- Doctor by training as he is, he has proved himself to be a successful writer.
I do not consider the article to be optional. "A doctor" is what he is. "Doctor" would be used only in direct address. Put it the other way around: Would you say, "He is doctor," or "He is a doctor?" I think only one of these could be correct.
Note that StoneyB is correct: "as" is used concessively only when it is comparative, as in "as old as he is" (StoneyB's example). The isolated "as" in your sentences do not constitute a concessive construction.
Might I also suggest a few constructions that I would consider more fluid, more pleasing, and more understandable? They are these:
"Even though he is a doctor by training, he has proved...."
"A doctor by training, he has nevertheless proved...."
"He is a doctor by training, and yet he has proved...."
The rest of the sentence deserves some discussion, too, but that's not something to go into here.