Council of Trent, Session VI, January 13, 1547, Decree on Justification:
"...By which words a description of the justification of the impious is indicated - as being a translation of that state wherein man is born a child of the first Adam to the state of grace and of the adoption of the sons of God through the second Adam, Jesus Christ, Our Saviour. 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".
Surely this affirms BOD and does not condemn it. The obvious sense of the words is not 'without the laver of regeneration and the desire', as you want to make it say, but one or the other, the laver of regeneration or the desire of the laver of regeneration.
Surely it does nothing of thie sort. "The wedding cannot take place without the bride or the groom." So we can have either the bride or ELSE the groom, and the wedding could take place?
If you try to make this into an either ... or type of statement, you have serious problems:
1) you claim that the laver of regeneration justifies without the desire (anathematized by Trent)
2) you elminate any notion of BoB that is distinct from and does not reduce to BoD (so the "three baptisms" become two)
3) if justification can happen with JUST the desire for it, the logical corollary is that justification can take place WITHOUT the laver ... which would be a denial of that same doctrine that Trent was dogmatically teaching, in other words, making Trent teach the same heresy it was trying to condemn
In order to teach what you claim it teaches here, Trent could easily have used an expresson like "vel saltem" (or at least) by desire, as it did for the Sacrament of Confession, or could have used the "vel ... vel" (either/or or both, whereas "aut" is either ... or but not both in positive expressions). Trent used both these expressions in its treatment of Confession.
In order to avoid interpreting Trent as teaching the heresy of #3 (that justification can take place WITHOUT the laver), the expression should have been, in Latin, "sine vel lavacro vel voto", using an inclusive or (rather than a disjuctive "aut" version of or) that does not teach that it can happen with the desire but without the laver.
There's a reason for the negative expression here, "Justification cannot happen without ..." It could have easily been made positive, along the lines of "Justification occurs by the laver or the desire," but this "Justification cannot happen without" implies necessary but not sufficient causes. If you were to say that "Justification happens by ...", you're not excluding that justification may also take place by other means. When you say that "Justification cannot happen without ..." you're saying that without these, it's not possible.
Trent was teaching here about the cooperation of free will with grace in the process of justification, against the Protestant errors, and if Trent were intending to teach about the "Three Baptisms," it surely would have mentioned "Baptism of Blood", which it nowhere does.