[NY] Anyone have experience in getting court name change order?

I'm in a bit of a predicament. So, my grandfather (who I'm using to get my citizenship) was never a naturalized citizen and to his knowledge he never "officially' changed his name. Basically, I'm trying to change his name from Joseph to Giuseppe on my mother's birth certificate (the discrepancy is my only obstacle to having all documents). The issue is the NYC department of health got back to me saying they needed to see the original supreme court order that changed his name. I don't know how to explain to them that he just decided to anglicize his name on the birth certificate and certain documents (for example, his marriage certificate reads Giuseppe and not Joseph) Plus his birth place isn't even correct on the birth certificate either. There was no petition or order. I simply just need a copy of my mother's birth certificate with Giuseppe as his name. I provided his original Italian birth certificate but that's not enough.

The issue is I dont want him to officially change his name in the courts because afaik his legal name is Giuseppe. I would love to contact a lawyer but I literally cant find the process or precedent for doing this. They said that the order needed to say that he wants to change his name from Joseph to Giuseppe on the birth certificate and it needs to include the child's name as well. I dont care about getting solicited at this point, if you're a lawyer who does this and doesn't have insane rates please contact me. Im at a dead end here.