Name Change for Minor of Non-US Citizen

My question involves name change laws in the State of: New Mexico

I was asked to research this for a friend of my wife. Here is the situation:
The mother is a Mexican citizen. She was visiting the US when she was 8 months into her pregnancy and ended up giving birth in the US. So, her daughter is a US citizen, with only the mother's last name. In Mexico, everyone's last name is a combination of the parent's last names. This is causing issues now that the girl is 10 years old.
I have found the forms for changing a child's name, but these are only for residents. What will need to be done here?
Thanks in advance.

Who wants to change the child's name, in what country, and for what reason?

Both parents want to change the child's name in the U.S. The child does not have a Mexican birth certificate, which creates issues. In order to get one, the U.S. birth certificate needs to be apostillized. But, this will result in a Mexican birth certificate with only one last name. Once again, creating big issues in Mexico. Getting a correction or change made in Mexico is next to imposible, so the only option is to get it changed in the U.S. first. Any assistance is greatly appreciated.