Your reading of Trent is completely wrong. You focus incorrectly on the word "or" without recognizing that it's in a double negative construction along with the preposition without. You need to read the passage in the entire context of Trent. I used to think Trent taught BoD and therefore believed in BoD (for catechumens). But I went back and read the ENTIRE Treatise on Justification in Latin, and it became very obvious that Trent was not teaching BoD.
Bill says that we cannot play baseball without a bat or a ball.
Out of context, and if you didn't know what baseball was, that could be ambiguous. Do you need BOTH or do you need ONE or the OTHER? This could be read as "We cannot play without (either a bat or a ball)"? (in Latin you would expect a double "or", an aut ... aut type of contruction before aut bat aut ball). But Trent doesn't use this construction. or else "We cannot play baseball without a bat or without a ball?" (meaning that you need both).
But what if I add the sentence:
Bill says that we cannot play baseball without a bat or a ball, since we need a bat and a ball to play baseball.
Immediately disambiguated.
Now look at Trent.
We cannot be justified without water or the will (votum = will, not just desire) for it, since Our Lord said that we cannot be born again without water AND the Holy Spirit.
Trent had just spent paragraphs discussing how the Holy Spirit acts in the soul to cause it to cooperate and to be properly disposed for the Sacrament.
Trent was teaching the relationship between the
ex opere operato nature of the Sacrament which however could only confer the grace of justification with the cooperation of the will. AGAINST THE PROTESTANT ERRORS. So Trent is making an analogy between the votum and Our Lord's reference to the Holy Spirit in the phrase "water and the Holy Spirit".
Trent is CLEARLY teaching that BOTH the water (Sacrament) AND the cooperation / proper disposition are required for justification. Without BOTH there is no justification.
Notice also the conspicuous absence of any mention regarding Baptism of Blood, which you would clearly expect if that's what Trent was actually intending to teach.
If you try to make the water or the desire thereof an "either ... or" proposition, then you turn the teaching of Trent into an ERROR. Why? Because you CANNOT BE JUSTIFIED WITH WATER WITHOUT THE WILL OR DISPOSITION. If you are not properly disposed and have the Sacrament performed on you, YOU ARE NOT JUSTIFIED.
Now let's look at some of the Canons in Trent:
Canon 3.
If anyone says that without the predisposing inspiration of the Holy Ghost[111] and without His help, man can believe, hope, love or be repentant as he ought,[112] so that the grace of justification may be bestowed upon him, let him be anathema.
Notice how this backs up my reading of Trent. Trent goes out of its way to say that the activity of the Holy Spirit to predispose the soul for justification is required, and to deny this is anathema.
Canon 4.
If anyone says that man's free will moved and aroused by God, by assenting to God's call and action, in no way cooperates toward disposing and preparing itself to obtain the grace of justification, that it cannot refuse its assent if it wishes, but that, as something inanimate, it does nothing whatever and is merely passive, let him be anathema.
Again, reinforcing the requirement of the will to cooperate in the grace of justification.
Canon 9.
If anyone says that the sinner is justified by faith alone,[114] meaning that nothing else is required to cooperate in order to obtain the grace of justification, and that it is not in any way necessary that he be prepared and disposed by the action of his own will, let him be anathema.
Again, ONE MORE emphasis on the need for the cooperation of the WILL (will comes from the same root word as "votum" in the famous passage that BoDers misinterpret).
Trent was teaching on Justification against the PROTESTANT ERRORS.
So when Trent teaches about not being able to be justified without the water or the will. It's not saying EITHER OR. In fact, it's emphasizing that the WATER (Sacrament working ex opere operato) REQUIRES ALSO THE COOPERATION OF THE WILL (="votum", usually translated wrongly as desire) towards justification.
It's ABSOLUTELY OBVIOUS that Trent is teaching that BOTH WATER AND (COOPERATION OF) THE WILL are required for justification.