It was actually quite lengthy so I'll sum up instead.
Transgenders are fine. Transgender children are fine. The choice belongs to the individual and not the parent. The parent has zero say. Zero. The child cannot be exposed to any treatments until they are adults.
It seems to me, in this instance, the child is lying to one parent.
If the child does believes themself to be transgender, and they are afraid what their father thinks, then the father is merely guilty for shaming his child. Which, hopefully, the child will be more willing to confront their father about this when they're older.
If the child doesn't believe himself to be transgender, and perhaps even claimed at one point for attention or simply no longer believes he is, then the mother is over-exaggerating the child's claims.
Either way, the mother is still wrong for trying to expose her child to treatment at such a young age. To me, it's up to the child to determine whether they are or are not transgender. Until then, they'll have to occasionally conform to their biological gender until society, hopefully, becomes more accepting and inclusive. But no one should tell them what to wear, how to talk, what to like, and most importantly - who they are, in the meanwhile.