The judge recognised that this was a minor infringement of Freddy's article 8 rights but concluded that it was a proportionate means of achieving a legitimate aim.
A child's right to an accurate record of its birth, and who gave birth to it, is a legitimate aim. So is a state's interest in keeping an accurate record of children born and their parentage, for all sorts of reasons.
Meeting these aims is clearly proportionate when weighed against Freddy's right not to be upset by the existence of a rarely viewed document that fails to validate Freddy's gender identity or legally fictional sex.
Especially as all of Freddy's actions have been about telling the world that Freddy is trans and pushing the GRA to its extreme limits.
Early in the case, Freddy applied for an order to keep F's identity secret. This was initially granted but later revoked because Freddy chose to televise the birth. Freddy's child had no choice about this. From the moment of birth this child has had all the same human rights as Freddy.
Incidentally, don't let anyone ever fool you that 'proportionate means of achieving a legitimate aim' is an exceptionally high bar for using the EA single sex exceptions. It runs through the Equality Act and through human rights law like 'Blackpool' through a stick of rock. Nowhere else is it held up as an exceptionally high bar. It's bread and butter stuff.