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Whereby the *Pelagians, and Calvinists be condemned, that promise life everlasting to young children that die without baptism, and all other that think only their faith to serve, or the external element of water superfluous or not necessary: our Saviour's words being plain and general.

So here, Rheims correctly upholds Catholic dogma and says that water is absolutely necessary.

Though in this case, God which hath not bound his grace, in respect of his own freedom, to any Sacrament,

Then Rheims contradicts himself and Trent by this statement, which is totally false.  God has, and does, bind his graces to sacraments.  Trent and Scripture infallibly tell us this.

may and doth accept them as baptized, which either are martyred before they could be baptized, or else depart this life with vow and desire to have that Sacrament, but by some remediless necessity could not obtain it. Lastly, it is proved that this Sacrament giveth grace ex opere operator, that is, of the work itself (which all Protestants deny) because it so breedeth our spiritual life in God, as our carnal birth giveth the life of the world.

If one reads the Church Fathers on baptism of blood, they consistently say that those who die for the Faith, are baptized by their own blood (and the angels baptize say the form and the blood takes the place of water as the matter of the sacrament).  "Baptism of blood" is nothing more than being "baptized by blood".  In other words, the Church Fathers understood BoB = baptism.  It was actual baptism.  It was a sacrament. 

The comments on "desire" being "accepted as baptized" is unsupported by the Church Fathers and Trent.  It's a theory, and arguably heretical because Trent tells us water is necessary.

Here's the same idea from Orestes Brownson:

So?  You can keep citing pro-BoD sources.  No amount of consensus will change the fact that Baptism of Desire is nothing but theological speculation that has been permitted and tolerated (wrongly IMO) by the Church.  BoD has not been revealed by God.

There are only two ways to discern whether something has been revealed, and BoD is an epic fail on both counts:

1) unanimous consensus of the Church Fathers -- yet only 2 Fathers (arguably and temporarily) floated the opinion, even while admitting it was speculation, whereas 5-6 Fathers explicitly rejected it.  No Church Father ever taught that BoD is doctrine received from the Apostles and taught it with authority.

2) when a truth derives logically and necessarily from revealed premises (where it's said to have been implicitly revealed) -- No such demonstration has ever been made.  Nearly every single theologian simply SAYS there's BoD but none of them ever prove it.  We have one alleged Scriptural "proof" offered, that of the good thief ... except the little detail that the Good Thief died in the Old Dispensation before the Sacrament was made obligatory after Our Lord's Resurrection.  St. Robert Bellarmine went with the opinion because the contrary "would seem too harsh" (the same theology held by 99% of all proponents of BoD, emotional wishful thinking).  St. Thomas made the only attempt to demonstrate it by syllogism, but it proved nothing.  He cited that the Sacraments have a visible and an invisible aspect, and then just stated that Baptism's invisible effects can be received by BoD.  CAN be and ARE ... these are two separate things.  No, the character, one of the two essential graces of the Sacrament, CANNOT be received invisibly.  God does not make "priests by desire" and confer Holy Orders that way.  There's an aspect of the Sacrament of Baptism that cannot be had without the visible reception of the Sacrament, so this proves nothing.  That's IT that I've seen by way of theological proof. 

Remaining theologians all rely on "Augustine and Ambrose" ... in ignorance.  St. Augustine temporarily floated the idea, admitting it was speculation, and that he went back and forth about it, and then retracted it later in life, issuing some of the most anti-BoD statements in existence at the end of his life.  St. Ambrose was speaking about a washing without crowning, but elsewhere he stated that even pious catechumens could not be saved if they died before actually receiving the Sacrament, so I submit that this Valentinian passage has been universally misinterpreted.  And what are the 5-6 Fathers who openly rejected BoD?  Chopped liver?  Most modern theologians just say, "Yup.  BoD".  "BoD".  Zero theological proof anywhere for this nonsense.

So there's no evidence anywhere proving BoD.  Consequently, it's nothing but sheer speculation (as admitted by Augustine).

On the contrary, the pernicious fruits of BoD are on display for all to see.  It's been used to absolutely gut EENS dogma, and if a concept of BoD had been condemned and not permitted, we could NOT have had Vatican II and the decay / corruption we see today.  Period.  This is why God allowed it, as a testing of the faith, and as the means by which He would introduce this final test of faith in the end times in which we live.