As far as a birth certificate, census or other legal document... in the US you would choose the nationality, not someone else. Whatever document it is there are choices (Caucasian, Black, Hispanic..etc) and you mark it, in fact in the US it is illegal to demand that kind of information in the first place. It is always optional if you want to list a nationality or ethnicity. None of my kids birth certs list a nationality or ethnicity.
I don't know anyone who would consider your baby black. I think it would depend on the person's appearance. My kids have many different nationalities of friends. They use the term 'black' as discriptive. For example: When we first moved into our home there were a lot of kids on the block that my boys began playing with and I had a hard time remembering their names. So my 7 yr old would say I am going to 'so and so's' house, the black kid or I am going to so and so's house, the blond kid - or the kid with glasses or the kid with the bird bath in front of their house... or whatever so that I would know which kid. So yes, if your child appears 'black' I could see them being seen as that descriptively. But at least everywhere that we have lived 'New York, Nevada and Colorado' people are people. My kids are the 'white' kids. If you want your child to know their heritage than teach it to them. I am Irish and so I make sure my kids know that they are too.
My one son is 11 and has never had a 'race' discussion with any of his friends. In fact about a month ago or so my son asked if his one friend who is hispanic was born in Africa because he has dark skin. He had heard the expression African-American and so it made him curious. That is when we had the discussion about peoples last names sometimes telling part of their heritage. His friends last name is Lopez so I would assume he is hispanic. The same as McLean or O'Reilly would be Irish. I personally don't like when people use African-American or Hispanic-American... I told my son that if you were born here or have citizenship here than you are American... period. But you have your ancestry like European or African or whatever. I tell them all the time that I am American but our ancestors, our heritage, is Irish. In all my experiences I don't think people would 'deny' your son his heritage. In fact... kids think it is super cool if you were born in another country. My son was real 'cool' because he was born in a different state and then a kid came into the class that was born in Spain and now he is the super cool one, lol.
I wouldn't worry about it. I really think it has a lot to do with the attention that the parent gives to it. Legally no one here can or would make you claim an ethnicity. Socially there may be assumptions but assuming someone is black is not a bad thing and your child should be proud of that, but if your child doesn't 'appear' European, then fill him/her in as they grow up.
I hope this helped some!!! Good Luck on your move in the coming years!!!