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Quote from: Ambrose

To deny Baptism of Desire is heresy.  For those that reject this de fide teaching of the Church, you place your soul in grave peril.

Is the de fide teaching limited to actual catechumens and martyrs?

Ambrose knows nothing but that "defide" line, which he repeats like a parrot.

Curiously, here's a BODer theologian quoted all the time by them, saying the opposite of what Ambrose is saying:

Dr. Ludwig Ott, Fundamentals of Catholic Dogma, The Necessity
of Baptism, p. 354: “1. Necessity of Baptism for Salvation;
Baptism by water (Baptismus Fluminis) is, since the
promulgation of the Gospel, necessary for all men without
exception, for salvation. (de fide.)”

Neil,

Can you add/make a syllogism with this:

Dr. Ludwig Ott, Fundamentals of Catholic Dogma, Membership in the Church, p.309:
“3. Among the members of the Church are not to be counted: a) The unbaptized… The so‐called blood Baptism and the Baptism of desire, it is true, replace Baptism by water in so far as the communication of grace is concerned, but do not effect incorporation into the Church… Catechumens are not to be counted among the members of the Church… The Church claims no jurisdiction over them (D 895). The Fathers draw a sharp line of separation between Catechumens and ‘the faithful.’”

But yet Ott and BODers believes that they are "kind of incorporated somehow" into the Church. By this statement, Dr. Ott is admitting that “baptism of desire” and “baptism of blood” are not compatible with Pope Eugene IV’s infallible definition on the absolute necessity of incorporation into the ecclesiastical Body (ecclesiastici corporis) for salvation.  

Pope Eugene IV, Council of Florence, “Cantate Domino,” 1441, ex cathedra: “The Holy Roman Church firmly believes, professes, and proclaims that none of those existing outside the Catholic Church, not only pagans, but also Jews, heretics and schismatics can become participants in eternal life, but they will depart ‘into everlasting fire which was prepared for the devil and his angels’ [Matt. 25:41], unless before the end of life they have been added to the flock; and that the unity of this ecclesiastical body (ecclesiastici
corporis) is so strong that only for those who abide in it are the sacraments of the Church of benefit for salvation, and do fasts, almsgiving, and other functions of piety and exercises of a Christian soldier produce eternal rewards. No one, whatever almsgiving he has practiced, even if he has shed blood for the name of Christ, can be saved, unless he has persevered within the bosom and unity of the Catholic Church                        

Fanciful in the extreme.

Does “baptism of desire” grant the grace of baptism/spiritual rebirth, yes or no

Yes, indeed it does. For spiritual regeneration or being born again is nothing other than that translation from the state of death to the state of grace that the Council of Trent speaks of, and this translation is effected by baptism or its desire.

First proof: Trent says, "And this translation, since the promulgation of the Gospel, cannot be effected, without the laver of regeneration, or the desire thereof, as it is written; unless a man be born again of water and the Holy Ghost, he cannot enter into the Kingdom of God."

Trent clearly teaches the sacramental effect of three sacraments - baptism, penance and the Holy Eucharist - can be received in fact or in desire, using the same word for each. To claim voto refers in two cases to the reception of the sacramental effect in desire and in the other case to a mere disposition is an illogical novelty that only the Dimond's and the unfortunate souls seduced by them can come up with. Trent further teaches that the sacrament of baptism is necessary just as the sacrament of penance is necessary - i.e. in fact or in desire.'

Some more proofs before the claim is answered.

Second Proof:

"...should any unforeseen accident make it impossible for adults to be washed in the salutary waters, their intention and determination to receive Baptism and their repentance for past sins, will avail them to grace and righteousness."

Third Proof:

Baptism, Necessity of Baptism and Obligations of the Baptized: 17 Q. Can the absence of Baptism be supplied in any other way? A. The absence of Baptism can be supplied by martyrdom, which is called Baptism of Blood, or by an act of perfect love of God, or of contrition, along with the desire, at least implicit, of Baptism, and this is called Baptism of Desire.

Fourth Proof:

Canon Law (1917): Canon 737: “Baptism, the door and foundation of the Sacraments, in fact or at least in desire necessary unto salvation for all, is not validly conferred except through the ablution of true and natural water with the prescribed form of words.”

All of which unanimously show that the traditional understanding of the Trent - the Church's own understanding - is that baptism is necessary in fact or in desire. No surprise, Doctors like St. Robert and St. Alphonsus, and every single authority and theologian besides explain the Tridentine teaching in this way.

Refutation of certain objections/misunderstandings in the OP:

1. Being born again requires the remission of all temporal punishment

False. Being born again, as seen above in Trent, is only the translation from the state of original or mortal sin to the state of grace. In this translation, it is not intrinsically necessary that temporal punishment be entirely remitted. The reason temporal punishment is entirely remitted in water baptism is owing to the work of Christ in the sacrament wherein it derives its especial efficacy. In the same way, attrition suffices in the confessional because of the special work of Christ in the actual sacrament, but contrition is necessary when the sacramental effect of penance is received outside the confessional.

As seen above in St. Pius X's Catechism, as explained by all Doctors, baptism of desire is nothing other than an act of perfect love of God or of contrition, in which desire for baptism is explicit or implicit, and therefore has a similar effect.

St. Thomas explains that contrition as an act of virtue works as a quasi material cause while as a part of a sacrament on the other hand it operates primarily in the manner of an instrumental cause, which explains the different effects associated with the same.

"Because, as part of a sacrament, it operates primarily as an instrument for the forgiveness of sin, as is evident with regard to the other sacraments (cf. Sent. iv, D, 1, 1, 4: III, 62, 1); while, as an act of virtue, it is the quasi-material cause of sin's forgiveness." (Supplementum Tertiæ Partis, Question 5 Article 1. Whether the forgiveness of sin is the effect of contrition?)

2. Why is only the temporal punishment removed?

Because of the nature of contrition as an act of virtue - it remits the eternal punishment entirely, but the temporal punishment only in part and in proportion to the intensity with which God is loved, as Trent itself explains when it is speaking of the nature of contrition, and also in its Catechism later - "not indeed for the eternal punishment which is remitted together with the guilt either by the sacrament or by the desire of receiving the sacrament, but for the temporal punishment which, as the Scriptures teach, is not always forgiven entirely." Thus, they will be saved, but only through purgatory. Therefore also, the Catechism of Trent points out that there is no danger for adult catechumens, because their determination to be baptized, and their repentance for past sins, will avail them to grace and righteousness. This being availed to grace and righteousness is what justification or being born again is, and Trent is very clear the same danger of death is not present for adults as it is for infants, therefore adult catechumens can be saved by baptism of desire.