"I am looking for a very good doctor for my child."
Translation:Çocuğum için çok iyi bir doktor arıyorum.
In this case you are right, because it is the object of "arıyorum".
But if it's the subject or predicate, then it would of course not get accusative case marking, definite or not.
For example: "O, iyi bir doktor." = He is a good doctor. "O, iyi doktor." = He is the good doctor.
As I understand it, the general rule is that nouns qualified by adjectives tend to be understood as definite unless an explicit "bir" is added, while nouns standing by themselves can be indefinite ("O, kitap okuyorum" = He is reading a book / He is reading books).
I don't think we ever allow answers like ""He is a good doctor = O iyi doktor" (if you have an example please tell me because it is wrong, and I need to edit it and warn the contributor who added it). If there is an adjective, you cannot omit "bir". However for "he is a doctor", it is more natural to say "o doktor", but "o bir doktor" is accepted.