Here's what I did step by step. I know you have covered a lot of this...but maybe this will help other people with the same problem. You may have done a lot of this, but I found backing up visits and phone calls with letters very helpful.
-called health dept, said I wanted to *register* a live birth at home, unattended. They gave me # of local vital statistics office. Ignore state, they just send copies of already registered births, as you know.
-talked to vital statistics gave her the details.
-heard nothing for a month
- requested legal copy of birth certificate from state office
- state vital statistics called us, they had no record of the birth being filed from the local office.
-called back local office and talked to supervisor of person I gave stats too
-faxed her a letter, too. Included names and numbers of people aware of the pregnancy. Mentioned we had photographs.

(I would use anyone, if you had no prenatal care. postal carrier, pharmacist, particularly if they hold any sort of job that requires a license. Though several adult legal residents of the same county would be close enough, if they confirm that they are aware you were pregnant and delivered. If baby does not have a ped, I would find one...like a general family practice doc, in new jersey you can probably find a Dr of osteopathy or someone else chill. Make a well baby visit or two. This is CYA stuff in case they decide to investigate you. Just having a dr takes a lot of that risk away.)
-she said the maternal-child dept would need to send someone to interview the mother. asked of the name of the supervisor there.
-talked to her, found out name of nurse that would be coming.
-faxed maternal-child supervisor similar letter to give to nurse
-called nurse to set time.
-at the same time, I called my local elected official...I live in a town so I called my city council person, who happens to be a lawyer. Discussed it with him. Outline the basic issue...birth at home being legal, refusal to grant BC to a birth that actually occured illegal. He said he though being in touch with our local county council person would be helpful. He was willing to make that introduction. Knowing how the departments inter-relate helps. In our county, the head of the health Dept also heads up the county vital statistics stuff. They must issue bc for live births that occur. That is their legal obligation. I would have had our county council person contact the head of the health dept to emphasize their legal obligation and maybe had a lawyer send a letter for me. Legal action is probably not required. The health dept's lawyer would probably suggest they issue after getting an letter from your lawyer. Most lawyers will write letters for you at pretty minimal cost.)
-nurse came. We had all witnesses - me, my mom, my sis the one who gave birth, and her DH available. Baby was present. Though her DH was not there at the interview in the end. Wanted witesses in case they started giving her a hard time. She said she would file paperwork immediately. Did not need to show her photos or the frozen placenta, but if she had been more hostile, suspicious, or wanted to do a physical exam, I would have. There was a midwife but she was lay, and therefore illegal in our state. We never mentioned her.
-one month later, called state vital stats. still no record.
-called orginal supervisor of vital stats. left voicemail. Interview done, nurse said she would file, what was happening?
-no return call, but magically 3-4 weeks later, confirmation card of record of birth files shows up in the mail. I know those papers were sitting on the supervisor's desk and she forgot to send them until my voicemail.
-Request copies of BC officially with payment. BC come a few weeks later.
Process took about 4 months. But I could have done this faster if I had followed up with more phone calls/letters.
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One other thing I would do, maybe. The Social Security Adinistration has some special processes for issuing SSN without a birth certificate. Many people use got SSN's when it first started did not have BC. Visit their web site, search. Call the national office, and your local office. Call or go see them. They may be able to exert pressure on the state for you. If end people you talk to don't know, have them move you up the chain.
another thing I considered. Call a hospital in your county that delivers babies. And you closest one that delivers a LOT of babies. Ask who files the BC paperwork. Some have a dedicated registrar. Be very nice to her...explain your problem. Ask for advice. Ask for names. See if she can make phone calls for you. Maybe bring the baby by to say hello. See if she'll help you file your paperwork. If she can't she may still know what to do. Every hospital has seen unexpected out of hospital births, "I had no idea I was pregnant"!, etc. so she'd know what to do. Call your closest birth center, repeat the process. Call anyone that does home births either family pracitioner or CNM, ask them, etc.
All I can say is, I know there are people who refuse to get BC and SSN's on purpose. It is MUCH easier to do it now than for your child to do it later. People that refuse to do so are making that decisions for the child that will follow them into adulthood and make things they might want to do (military, professional careers, driver's license, marriage, travel outside the US) very difficult. So it's just something to think about when deciding to forgo any documentation at all. But there are definitely two sides to that argument, there are many people who feel very strongly about not doing so.