Can't see how you can be stopped. Not sure why anyone says that a church funeral is not a public event.Never heard of the public being barred from one,nor of anyone being issued with an invitation to one, bar one of a great public figure like the Duke of Wellington.No doubt it is possible to have some part private, such as any related communion,but not
the whole event surely?
Anyway, barring one person, the son of the deceased, alone is hardly a Christian thing to do, is it?
Even Church weddings are, strictly speaking, public events, but that's perhaps because of the need to have them witnessed by the congregation at large and to meet the requirement that any person knowing of any 'lawful impediment' be present to raise it. In practice they are arranged so family and close friends have priority over the church seating and are invited, but they are public