If it's a public school, you don't need proof of legal residency to enroll a child. You can go through the enrollment process quickly when you get here because all children are required to attend school up until the age of 16. Many undocumented children have no problem going to school and usually hit a road block when it comes time to apply for college, but that's not your case.
"All children in the United States are entitled to equal access to a public elementary and secondary education, regardless of their or their parents’ actual or perceived national origin, citizenship, or immigration status. This includes recently arrived unaccompanied children, who are in immigration proceedings while residing in local communities with a parent, family member, or other appropriate adult sponsor."
https://www.ed.gov/sites/ed/files/policy/rights/guid/unaccompanied-children.pdf