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If you actually believe this, then this belief hinges on the impossibility for the person to receive the sacrament.

I ask again for you to come up with a circuмstance where it is impossible for God to provide the sacrament to the sincere soul about to die.

You are incorrect.  The teaching is not based on the idea that it is impossible for God to provide the sacrament to the sincere soul about to die.  

But in a sense He does, so long as that soul in addition to being sincere has a supernatural faith and perfect charity.  The sincere soul as you know, is not aware of the necessity of baptism or the necessity of belonging to the Church through no fault of his own (how many such people are there?). This soul is baptized by the Holy Ghost in that such a one is cleansed of Original Sin by Him.  This baptism of the Holy Ghost is what is referred to as "Baptism of Desire".  

The authoritative Docuмents and the infallible ones teach baptism of desire being fully aware that God can baptize with water the sincere soul about to die.  They are also aware that He does not always do so and is not obliged to do so, but in fact that He rarely works it out that they get sacramentally baptized through some miracle.  But the innocent does not get condemned on this account because his soul is cleansed by sanctifying grace and not by water in these circuмstances.  

You are putting the apple before the cart insisting on that which is necessary with a relative necessity of means (sacramental baptism) over that which is necessary with an absolute or intrinsic necessity of means (sanctifying grace).  This hypothetical man we speak about is saved within the Church through the intrinsically necessary means.   

How does one have such virtues when unbaptised?  If they don't know about the Church, what do they have supernatural faith in?  If they don't know about the Trinity, or of Christ's death on the cross, how do they have PERFECT charity, seeing as they don't know anything about God?

How can a soul have supernatural anything, if it's in the state of original sin?  And probably mortal.  A soul in sin does not have supernatural grace!!!!!!!

Again.  You ask this as if all the theologians, Fathers, Saints, Doctors and Popes who teach this are inventing the impossible when they teach this.  Supernatural Faith is a gift freely given.  It is given to those of good will (not those who seem to be of good will but the few who actually are of good will) through the Catholic Church or through some miracle.  God can do this.  He is not dependent on water to do this.  This is what the Church teaches.  Remember Trent speaks of both Baptism and Confession as being fulfilled by desire if receiving the Sacrament itself is impossible.  These are the only two because the first is necessary for salvation to be possible and the second necessary for one who has committed mortal sin.  Confirmation, Orders, Marriage and Last Rites are not necessary for salvation.  One can be saved even apart from the Eucharist, I'm sure you are aware of when this is possible.  The Eucharist can also be "desired" spiritually received.  This of course is not the same as receiving It sacramentally much as BOD is does not give one the indelible mark that the Sacrament does.  But in the later case it does what is necessary for salvation as does perfect contrition when the Sacrament of Confession is not possible.  God is perfectly just and merciful.  One is condemned to the pain of sense in Hell for ever only when one is guilty of mortal sin.

It is absolutely impossible for a soul who does not have supernatural faith i.e. belief in God based upon God revealing (and this is ordinarily done through the Catholic Church) if not from the Catholic Church then it would take a miracle like what happened to Saint Paul.