Not all fares are actually listed on BR Fares, some can also be missing in theory.
Maybe ask your local ticket office about the flat £2 child (if an adult accompanies them) fare since that is the only place that offers those.
Also, if your child is travelling on a 5-10 Zip Oyster, they may be entitled to free travel on that journey nonetheless.
TfL's free travel scheme doesn't just apply to TfL services and NR services where LU tickets are recognised, but also some other NR services (incl. some or all operated by GTR excl. GX) as well.
