Harry is now in Year 6, his final year of Primary School. The FSSP priest cannot put Harry in a normal RCIC class: 1) Harry is a nephil, not a human 2) the Dursleys will not consent, 3) he is no more successful at engaging the local ACPCs1 than Harry has been. He is educating Harry on Catholicism and about the magical world, but from the Church’s perspective. The Church never signed the International Statue of Wizarding Secrecy (ISWS), but does mostly tacitly obey it and keep things quiet/secret. As a result though, the Church knows far more about the magical world than anyone magical would suspect, in some cases more than they know about themselves.
For the story to work, Harry must not be baptised until sometime after he starts Hogwarts. This can thus cover Year 6, but no formal conversion. Like Chesterton, Harry must delay his conversion for some time after he has converted in his thinking.