If neither you or your husband are the biological parents then I would think that you would have to adopt her. If there is no father on the birth certificate and the birth mother's parental rights have been legally terminated adoptions should not be a problem. Adoption would provide much more security for her and for you.
Actually, the only way you can change a name on a birth certificate is thru adoption. And......unless your name is on the birth certificate you cannot obtain a certified copy of her birth certificate. This may be a problem for her in the future.
Since she is underage I doubt that you can legally change her name in any way other than thru adoption.
You can use your family's last name on everything. I suppose you are to use a child's legal last name on school records but if this is a law it is not enforced. My daughter changed her daughter's last name to a hyphenated one when she was married. When she was divorced she changed her own name back and dropped the last part of her daughter's hyphenated name. The school said that they would use whatever name they were given. They do not check court records or require a court record to change the school records.
I adopted my daughter who came to me as a foster child. I knew other foster children who used their foster parent's last name in school once they had been permanently placed with that family for the purpose of adoption but the adoption was not as yet applied for.
As a police officer, I've seen many children and adults using last names different than the ones on their birth certificates. This does cause a bit of confusion sometimes but once the person consistently uses the same last name it becomes accepted as if it were legal. If you or your daughter, when she's an adult, needs records such as her birth certificate you/she will just have to use the legal last name. For some purposes you/she can use the term aka, also known as.
I know of no one who checks to see if last names used are legal last names. The police certainly do not. My suggestion is to drop her birth name and use your family name if the name is your only concern.
Not knowing her history, I can only say it seems that adoption would be the best way to change her name so that she is legally your child. It is possible that the state could challenge your right to have the child under certain circumstances. Not because she doesn't have your last name but because neither of your names are on the birth certificate. Since her birth mother's parental rights have been legally terminated she cannot have given permission for your husband to have custody. I could see the potential for this to be a legally tricky situation.
I would contact an attorney and ask where you and your daughter stand legally. You would not have to say that she is not your husband's child. His name is not on the birth certificate and needs to be put on the birth certificate. This is the only way I know for this to happen but there may be other ways.
I know nothing about legal guardianship. If you have a legal document granting you guardianship then you may be able to do certain things such as get a certified copy of her birth certificate. You may also be able to legally change her name but you can't change her birth certificate.
