The infant baptized outside the Catholic Church is not a member of the Church. These children are members of the schismatic and heretical sects into which they have been initiated.

Were they cleansed of original sin at baptism? Yes, provded that the form and matter of the baptism are correct.

Were such an infant to die, would that child enter eternal beatitude? Whilst many Catholic churchmen in most recent times driven by sentiments of good intention and emotional hope have answered in the affirmative, the Church has definitively taught otherwise in its dogmatic starement on salvation from the Council of Florence. These children were they to die would be damned. They formally belong to a sect outside the Church.

However, the child is not and never has been a member of the Catholic Church, membership in which is a necessary prerequisite for salvation.

Should they die, they will be damned. The severity or mildness of their punishment is known to God alone until the General Judgment, but they would be eternally excluded from the Beatific Vision.

I don't believe parental authority extends/encompasses a mere declaration (welcome into their church/coven etc) that would result in their child being in any state of schism/heresy before the use of reason.  I don't believe any parent/guardian has the authority to impart a sin, venial or mortal, on their child anymore than anyone could transfer their schism/heresy to another without their co-operation.

The infant baptized outside the Catholic Church is not a member of the Church. These children are members of the schismatic and heretical sects into which they have been initiated.

Were they cleansed of original sin at baptism? Yes, provded that the form and matter of the baptism are correct.

Were such an infant to die, would that child enter eternal beatitude? Whilst many Catholic churchmen in most recent times driven by sentiments of good intention and emotional hope have answered in the affirmative, the Church has definitively taught otherwise in its dogmatic starement on salvation from the Council of Florence. These children were they to die would be damned. They formally belong to a sect outside the Church.

"The effect of this sacrament [of baptism] is the remission of all original and actual guilt, also of all penalty that is owed for that guilt. Hence no satisfaction for past sins is to be imposed on the baptized, but those who die before they incur any guilt go straight to the kingdom of heaven and the vision of God." - Council of Florence

I don't know what ER is talking about.