you are not committing any fraud. on the application, just write the birth certificate number and date. as long as you have the certificate to prove that the data is real, you shouldn't worry. parents' status don't usually come up during the passport application.
i say, go and try for passport application. give the data as honestly as you can. then see what reply you get from the officer. if you are rejected, then you know for sure that this process won't work. better than living in dilemma i suppose. all the best.