My mom had a problem with her name because there was "Maria" in her birth certificate that she did not know all along, she just went to a lawyer to have an affidavit that she is the same person, once you have that affidavit you can apply for a passport using the name in the NSO birth certificate, it should not take that long. One day for the affidavit, then applying for a passport - that I am not sure maybe pay for rush processing. However get some documents ready to backup previous names used like secondary Form 137, baptismal certificates and government ID's. It should be fine, it is doable. Make sure also to make photocopies of the affidavit, colored scanned copies would be better, the originals will not be returned to you when submitted for passport application, the immigration will take photocopies of the affidavit, you will need to have an explanation letter of the name change and attach the affidavit with that. You can just write a simple one indicating that it was a clerical error and you were advised to follow what is written on the birth certificate. Whenever you will be asked for an NBI clearance for your brother I am sure they will require AKA of the other name used, just like my mom's, however it is easy to do, the main office does that just make sure you fill out the online application with the AKA that the immigration asked and it will be printed with that...