From the school's perspective, the only address that matters for establishing a residency location is that of your girlfriend as the child's legal parent.
If none of your girlfriend's documents reference your ex's address (which I am assuming they don't), then there will be no documents to prove the child lives in your ex's school district zone. Your girlfriend would need a driver's license showing the Hough High address, vehicle registration, utility statement, bank statement, et... that proves she lives in the Hough High zone.
The list of required documents to prove residency is here:
https://www.cms.k12.nc.us/cmsdepartm...%20English.pdf
For school transfer, there is currently no space at Bailey and Hough. You can apply next year for School Choice though. There is not a category though for grouping a girlfriend's child that you are not a guardian for with your own legal children as a unique hardship. Preference is possible for children with the same legal parents.:
https://docs.google.com/spreadsheets...t#gid=40840822