Hi thank for your advice.I'm not an expert on this, but in parallel I'd consult with a lawyer who may be able to suggest something. I don't know what exactly - but eg a legal opinion stating that if there is no mother on the birth certificate, that you are by law the only parent, and any biological mother (if ever surfaces) would have no parental rights. And no bar on leaving the country with the child (as you are the only parent).
An opinion like this in context of the comments above (eg that a birth certificate without a mother is rare but occurs due to social pressures, or something like that).
That may not be better than the other options you're pursuing, but useful if those do not bear fruit.
I don't know what kind of lawyer would handle this. Probably someone who handles adoptions and perhaps has handled international adoptions.
I have consulted several lawyers in China: blood relationship is regarded by court the most important factor for parentship. Normally, both bio-parents get custody right of the child automatically, regardless missing parent info on birth certficiate (as long as you can prove blood relation with the child). Besides, the best intestest of the child is considered. If one parent abuses the child, gets drug addicted or mentally disabled etc., the court will deprive the custody right, and give it to the other parent, but only in some extreme cases.
So based on Chinese law, the bio-mother should have the custody right. I know it's the best to get the birth certificate corrected with the bio-mother, but in reality it seems it's impossible.
If I can get the real name and ID number of the bio-mother, I guess I could go to court and get a court decree clearly stating the custodians of the child? Then I can get the signature from the mother.