CathInfo.com

But either way, to say as the quote says: "no one is saved without the laver of regeneration or the desire for it," is to say no one is saved without the sacrament. To add "or the desire for it" is to say neither is anyone saved with the desire for the sacrament.

Right. It's the same as if one were talking about the Sacrament of matrimony, where there can be no marriage unless there is bridegroom or bride. Can you have marriage with a bridegroom only? Or a bride only? No, both are required. It comes right back to the core issue here: with baptism, you cannot be saved with the laver of regeneration but no desire for it. And vice versa, you cannot be saved with a desire for it but no laver of regeneration.

That's why I postulate that BOD and BOB are both accidental causes of the justification already formally introduced by the actual Sacrament of water baptism. BOD is really nothing more than perfect contrition, and BOB is nothing more than Holy martyrdom. Both have a similar effect as baptism: they justify and remit sin by the merit of their action. But you must have already been baptized for it to have any effect at all.

Pope Eugene IV, “Cantate Domino,” Council of Florence, ex cathedra:  “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.”

(My emphasis)

He also speaks of "justified" in paragraph 13, so he even seems to contradict himself:

There are some other problems with St. Alphonsus' thinking.  He holds that if someone is saved by BoD, then temporal punishment for sin can still remain.

But that contradicts what Pope Innocent taught on the matter (so, in holding such letter to be de fide), that would make his opinion there heretical.  That Pope claimed that the one individual who he confidently "asserts" is in Heaven (not just that he COULD have been saved), but would go to heaven immediately and without delay.

Also, Trent clearly teaches that initial justification is a rebirth or regeneration, and then defines rebirth/regeneration as a COMPLETE renewal of the individual, wiping out all traces of sin or punishment due to sin.  This was highlighted by the Dimond Brothers, and they are quite correct.

BoD theory is fraught with contradiction and confusion, with wide ranges of theories about what it does, what it doesn't do, etc. etc.  To me that's a clear indication that it's never been taught by the Church.  You cannot believe in something without knowing what that something is.  I can't believe in a phrase "BoD", but can only assent intellectually to various propositions.