The relevant question is how did Baptism of Desire get extended to followers of false religions? You won't find St. Alphonsus teaching that. His interpretation of "de fide" doesn't include Hindus, Jєωs, Muslims, etc.

The relevant question is: Why would a Catholic quote Fathers, Doctors, or generally Saints like St. Alphonsus or St. Thomas Aquinas to learn about the true Faith, while the competent authority on Faith is the infallible Magisterium of the Church?

The ecuмenical Vatican Council makes all teachings of the ecuмenical Council of Trent with respect to justification part of the Profession of the Faith of the holy Roman Church (see here). The ecuмenical Council of Trent does not teach any "Baptism of Desire" (not even for a catechumen) and strictly forbids to teach, preach, or believe any such thing (see here).

The ecuмenical Council of Trent teaches (source see here):

Of this Justification the causes are these: the final cause indeed is the glory of God and of Jesus Christ, and life everlasting; while the efficient cause is a merciful God who washes and sanctifies gratuitously, signing, and anointing with the holy Spirit of promise, who is the pledge of our inheritance; but the meritorious cause is His most beloved only-begotten, our Lord Jesus Christ, who, when we were enemies, for the exceeding charity wherewith he loved us, merited Justification for us by His most holy Passion on the wood of the cross, and made satisfaction for us unto God the Father; the instrumental cause is the sacrament of baptism, which is the sacrament of faith, without which (faith) no man was ever justified; lastly, the alone formal cause is the justice of God, not that whereby He Himself is just, but that whereby He maketh us just, that, to wit, with which we being endowed by Him, are renewed in the spirit of our mind, and we are not only reputed, but are truly called, and are, just, receiving justice within us, each one according to his own measure, which the Holy Ghost distributes to every one as He wills, and according to each one’s proper disposition and co-operation. For, although no one can be just, but he to whom the merits of the Passion of our Lord Jesus Christ are communicated, yet is this done in the said justification of the impious, when by the merit of that same most holy Passion, the charity of God is poured forth, by the Holy Spirit, in the hearts of those that are justified, and is inherent therein: whence, man, through Jesus Christ, in whom he is ingrafted, receives, in the said justification, together with the remission of sins, all these (gifts) infused at once, faith, hope, and charity. For faith, unless hope and charity be added thereto, neither unites man perfectly with Christ, nor makes him a living member of His body. For which reason it is most truly said, that Faith without works is dead and profitless; and, In Christ Jesus neither circuмcision, availeth anything, nor uncircuмcision, but faith which worketh by charity. This faith, Catechumen’s beg of the Church-agreeably to a tradition of the apostles-previously to the sacrament of Baptism; when they beg for the faith which bestows life everlasting, which, without hope and charity, faith cannot bestow: whence also do they immediately hear that word of Christ; If thou wilt enter into life, keep the commandments. Wherefore, when receiving true and Christian justice, they are bidden, immediately on being born again, to preserve it pure and spotless, as the first robe given them through Jesus Christ in lieu of that which Adam, by his disobedience, lost for himself and for us, that so they may bear it before the judgment-seat of our Lord Jesus Christ, and may have life everlasting.

There is no "Baptism of Desire", not even for catechumens. A catechumen cannot have the Faith. The Faith is infused when receiving the sacrament of baptism.

Noone has "supernatural Faith" or Faith at all, if Faith hasn't been infused when receiving the sacrament of baptism. That's what the competent authority on Faith, the infallible Magisterium of the Church, teaches. That's what the Council of Trent teaches, and what the Vatican Council declares to be part of the Profession of Faith of the holy Roman Church (see here).

Please, stop quoting fallible Fathers, Doctors, and Saints, on questions which have been infallibly answered, defined, and decreed by the infallible extraordinary Magisterium of the Church of our Lord.

I prefer to confess the Profession of Faith of the holy Roman Church here and now, and I prefer to die confessing the Profession of Faith of the holy Roman Church.

Well Trent says faith comes from hearing. So even if you want to throw out the ordinary magisterium, Struthio, Trent isn't going to get you to BoD denial. Certainly not the way you just described, at any rate.

Well Trent says faith comes from hearing. 

What a Protestant sounding attempt at a rebuttal.  Well done, brother Mith!  

Well Trent says faith comes from hearing. So even if you want to throw out the ordinary magisterium, Struthio, Trent isn't going to get you to BoD denial. Certainly not the way you just described, at any rate.

Oh-oh...

Well let's take one thing at a time. I'm happy to debate the point directly from Trent, so long as I'm also on record noting that the real problem here is a fundamental misunderstanding of Catholic learning viz. disregard for the ordinary magisterium.

When describing the justification of the sinner before baptism, Trent teaches (contra the Protestants) the initiatory role that faith plays in that process. It says faith comes from hearing, not baptism. Denz 798

"Now they are disposed to that justice [can. 7 and 9] when, aroused and assisted by divine grace, receiving faith "by hearing" [Rom. 10:17], they are freely moved toward God, believing that to be true which has been divinely revealed and promised..."
.
It says the virtues are infused at baptism too, of course, which only makes sense given the (again, contra Protestants) *continued* justification of souls.

But that isn't what it says, Struthio. It says they receive faith by hearing. It describes that faith as a *grace* whereby man believes all that God has revealed. It then describes the acquisition of the other two virtues (hope and charity). All before baptism.

It says that "they are disposed to that justice". You need to be disposed to receive justification by the sacrament of baptism. Hearing does not transmit the faith, hearing disposes to be prepared to receive the faith.

.
It seems to me that there are some who, to defend BoD as being "de fide" or even "ex cathedra" (seriously!), are coming from a place of ignorance regarding the nature of grace. They confuse actual grace with sanctifying grace, and I have met a few who claim there is no distinction, that all graces are the same; such that pagans ignorantly worshiping at a pagan shrine are thereby recipients of the same sanctifying grace (and therefore justification and salvation without need for Baptism) that the Apostles received on Pentecost Sunday.
.
You can't make this stuff up!
.
When did the Apostles teach that receiving the Holy Ghost on Pentecost made them all exempt from having to be baptized?