You obviously didn't read what I quoted: "In like manner, suppose a person living in a false religion dies without giving any sign of embracing the true faith, or without being reconciled to the Church of Christ, we can never say of such an one with certainty that he is lost; all that we can say must be under the same condition as in the other case: if he has actually died as he lived, separated from the true Church of Christ and without the true faith of Christ, he cannot be saved. But if God, of His great mercy, has given him in his last moments light and grace [as Pope Bl. Pius IX also taught] to see and embrace the true faith, AND he has corresponded with so great a favor as God requires, he will be saved...." This was Bp. Hay's Catechism. Fr. Mueller's, which you quoted me quoting above, teaches the same thing. How is it so many approved Catechisms teach the same thing if they are all wrong? It doesn't mean anyone is automatically saved. It just means that, by an extraordinary miracle, in response to the prayers of the Church, God may choose to save some.
The Church prays daily for souls near death to be saved, for the just to gain perseverance, for sinners to be converted. Her Priests and Her faithful offer up many prayers and sacrifices. Not all we desire to save will be saved, but some will be.
God will not completely reject the prayers of His Church and Her faithful to save souls, especially when they make sacrifices for it, as Our Lady of Fatima taught. It's not our duty to pronounce final judgment on anyone, but to pray for all.
God alone knows with certainty how He will apportion the graces of our prayers. Some will respond to Grace and convert. Some will not.