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So the Church, or the godparents, provide what is lacking in the infant (the vow, the desire),

Yes, because the Church tells us this is allowable.

but the Holy Ghost can't supply for a lack of water?

The Church has never said this is possible.

The Scotist theologian Claude Frassen (d.1711) who was post-Trent, and an expert in the writings of Blessed Duns Scotus, devoted a whole chapter in his tenth volume to the question of the necessity of baptism (De necessitate Baptismi).  Without giving the whole article, I translate his first conclusion:

"Baptism is necessary by a necessity of means for all men, whether adults or children.  This is determined de fide from the Councils of Milan and Carthage, as Augustine puts forth, Epistle 90, and 92, and from the Council of Trent, Session 7, Can 5., 'If any one will have said, that Baptism is optional, that is, not necessary for salvation, let him be anathema.

But the truth of this conclusion is gathered from the words of Christ in John Chapter 3, 'Unless you will have been born of water,' etc. By such words Christ the Lord signifies, after the promulgation of the Law, that no man is able to gain eternal salvation, unless he will have been born by the water of Baptism.  Such is the teaching as laid forth in Tertullian in his book concerning Baptism, ch. 12, where he says, 'It is prescribed, that no man is able to achieve salvation without Baptism, from this, primarily from the words of Our Lord, 'unless you will have been born again from water, you will have no life.'  Hence the chapter following he (Lord) is speaking concerning those who assert that faith suffices for salvation...By these words (Frassen continues) Tertullian very patently reveals the impious assertion of the Calvinists, who contend that man is able to be justified, even without baptism."

Later in the article Frassen dissects the words "unless a man be born of water and the Holy Ghost," commenting that all the fathers (omnes SS. Patres) take the conjunction 'and' not as a disjunctive, but as a copulative word, meaning that both faith and water are necessary.  

Another serious problem with the BoDer rendering.

Since justification cannot happen without actual Baptism or the Baptism of Desire, then what of Baptism of Blood?

If Trent was teaching here about the alleged "Three Baptisms", why no mention of BoB?

If justification cannot happen without the Sacrament or the desire, then BoB ceases to exist as an independent thing, and we only have 2 Baptisms.  BoB must reduce to a form of BoB ... but no BoD theorists hold that.  To claim that there's a separate BoB that can justify and save would therefore be heretical by your own criteria.

St. Alphonsus' theory about a BoB that works "quasi ex opere operato" and has the additional effect (vs. BoD) of remitting temporal punishment due to sin, would be heretical, since it's ruled out by this passage in Trent.

ANSWER:  Trent was NOT teaching about the so-called 3 Baptisms at all, but rather, against the Protestant errors, affirming that both the ex opere operato grace of the Sacrament TOGETHER WITH the cooperation of the free will (the votum) are required for justification.  You'll note that the word "votum" in Latin is derived from the world "to will".

To say that justification can happen without the Sacrament is to effectively render salvation ex opere operantis (which is Pelagian heresy).