In Catholicism, does the efficacy of the sacrament depend on the intention of the priest?

This issue of intention had come up in 1896 as regards the Catholic view that Anglican orders were null and void. This was so because their "form" and "intention" were wholly different from Catholicism's.

33. With this inherent defect of "form" is joined the defect of "intention" which is equally essential to the Sacrament. ... if the rite be changed, with the manifest intention of introducing another rite not approved by the Church and of rejecting what the Church does, and what, by the institution of Christ, belongs to the nature of the Sacrament, then it is clear that not only is the necessary intention wanting to the Sacrament, but that the intention is adverse to and destructive of the Sacrament.

34. Wherefore, strictly adhering, in this matter, to the decrees of the pontiffs, our predecessors, and confirming them most fully, and, as it were, renewing them by our authority, of our own initiative and certain knowledge, we pronounce and declare that ordinations carried out according to the Anglican rite have been, and are, absolutely null and utterly void. - Apostolicae Curae On the Nullity of Anglican Orders

So yes, the Catholic Church teaches that the efficacy of their Sacraments depends on intent.

The Catechism of the Catholic Church (CCC 1256) may shed some light on this:

The ordinary ministers of Baptism are the bishop and priest and, in the Latin Church, also the deacon. In case of necessity, anyone, even a non-baptized person, with the required intention, can baptize, by using the Trinitarian baptismal formula. The intention required is to will to do what the Church does when she baptizes. The Church finds the reason for this possibility in the universal saving will of God and the necessity of Baptism for salvation. (emphasis mine)

There is a distinction between the intention of the minister and his holiness or sinfulness. The Church is clear that a sinful man (aren't we all?) can effect valid sacraments. But if he doesn't intend to effect a sacrament, that's a different matter. Personally I think that clause in the catechism is really just creating an exception for, for example, actors doing a baptism scene in a movie. But if I learned my children were baptized by the faithless minister, I'd still do a second, conditional, baptism just to be sure.

The Church's current teaching points to the sacrament itself overcoming any weakness of an individual.

If a priest intends to send babies to hell while he is baptizing them, are those infants deprived of regeneration, according to Catholicism?

No. The presumption made that a priest intends to send babies to hell is based on absurdity (addressed later) similar to "if Korvin placed his hand in a blender with the intention of trimming his nails ..."

1. Canon law makes the presumption that a sacrament is entered into with correct intent. (An exception must be demonstrated)
2. Extraordinary claims require investigation. The claim that an ordained bishop/priest/deacon did as described requires proof to be credible: such a bizarre case would need to be investigated by a tribunal. For example, the sacrament of matrimony accrues the favor of the law unless it is asserted that the marriage was not sacramental. In such case, a tribunal must examine the evidence and forward its findings on the facts of the matter to the local ordinary, and now to the archdioceses.

CCC 1128 This is the meaning of the Church's affirmation⁴⁹ that the sacraments act ex opere operato (literally: "by the very fact of the action's being performed"), i.e., by virtue of the saving work of Christ, accomplished once for all. It follows that "the sacrament is not wrought by the righteousness of either the celebrant or the recipient, but by the power of God." ⁵⁰ From the moment that a sacrament is celebrated in accordance with the intention of the Church, the power of Christ and his Spirit acts in and through it, independently of the personal holiness of the minister. Nevertheless, the fruits of the sacraments also depend on the disposition of the one who receives them.

You ask in a follow up comment:

What does "From the moment that a sacrament is celebrated in accordance with the intention of the Church" imply? If the priest's intention is not in accord with the Church's, does this section apply?

I respond:
The Church is the Body of Christ and is bigger than any one person in it or any one person serving it.
The intention of the Church is that all be saved through Baptism. (Great Commission, etc)
That's how I heard it explained, and is consistent with how the church generally see the sacraments. In the modern vernacular, if "on a given day" a priest does not have his A-game" the Church always does. Why? Because the Church is (a) the Body of Christ, and (b) One Holy Catholic and Apostolic Church. The holiness of the church is not abrogated by the sinfulness of one member.

Problem's with Clark's polemic

Assuming (1) a priest intends to send babies to hell, and (2) said priest with that intention would not first go to confession to cleanse himself of evil and lack of charity -- but instead with hate in his heart perform a baptism -- is to create an argument of the absurd. Give me a freakin' break. Whence comes this assumption? From a lack of Christian charity. While I question the honesty, the intent, and motive of anyone proposing that absurdity (Clark) we can play along since the polemic is now in play.

By having evil intention, per Clark's absurdist example, the clergyman by abdicating Christian charity falls into the Donatist category (Canon XII from Trent, which follows Canon XI, the basis of the argumentum absurdum) that you asked to set aside in establishing the criteria for an answer. (Why would one suppose things happen in a vacuum?)

The Council of Trent covered more than one base in the Canons. In any case, the current Church has addressed this in the cited CCC 1128.

The above considered, @MattGutting pointed to Thomas Aquinas' Summa Theologoca, III, Q64 which is one of the notes for the CCC; it finds Thomas arguing with himself on this edge case.

Reply to Objection 2: On this point there are two opinions. For some hold that the mental intention of the minister is necessary; in the absence of which the sacrament is invalid: and that this defect in the case of children who have not the intention of approaching the sacrament, is made good by Christ, Who baptizes inwardly: whereas in adults, who have that intention, this defect is made good by their faith and devotion.
Consequently, others with better reason hold that the minister of a sacrament acts in the person of the whole Church, whose minister he is; while in the words uttered by him, the intention of the Church is expressed; and that this suffices for the validity of the sacrament, except the contrary be expressed on the part either of the minister or of the recipient of the sacrament. (Further up, Aquinas posits "intent" thusly: And this intention is expressed by the words which are pronounced in the sacraments; for instance the words, "I baptize thee in the name of the Father," etc")

One can read this as "if he's up there doing the baptism, and says the words, the Church takes over." (Or one can choose not to). Part of the citation goes directly to personal intention being an obstacle, while the other passage argues the opposite.

Given that the CCC citation appears to be a product of synthesis over the years, the conclusion in the CCC stands up when the juxtaposition of both Canons XI and XII of Trent are considered.

I added citations from the CCC's notes. I'd recommend a review of CCC 1117-1129 (if you can handle the corpulent prose) to get the larger view of how the Church sees the sacraments. From the proceedings of Trent, it seems a mistake of reasoning to set aside the Donatism in relationship to the other matter. Clark's attack looks like both a case of special pleading and an exercise in the absurd.

⁴⁹ Cf. Council of Trent (1547): DS 1608.

CANON XI.-If any one saith, that, in ministers, when they effect, and confer the sacraments, there is not required the intention at least of doing what the Church does; let him be anathema.
CANON XII.-If any one saith, that a minister, being in mortal sin,-if so be that he observe all the essentials which belong to the effecting, or conferring of, the sacrament,-neither effects, nor confers the sacrament; let him be anathema.

⁵⁰ St. Thomas Aquinas, STh III, 68, 8.

Regarding CCC note DS 1608; Sources Catholic Doctrine (Denzinger-Schonmetzer; Enchiridion Symbolorum, definitionum et declarationum de rebus fidei et morum):

The Ordinations of Schismatics * [From the epistle (1) "Exordium Pontificatus mei" to Anastasius Augustus, 496] 169 (7) According to the most sacred custom of the Catholic Church, let the heart of your serenity acknowledge that no share in the injury from the name of Acacius should attach to any of these whom Acacius the schismatic bishop has baptized, or to any whom he has ordained priests or levites according to the canons, lest perchance the grace of the sacrament seem less powerful when conferred by an unjust [person]. . . . For if the rays of that visible sun are not stained by contact with any Pollution when they pass over the foulest places, much less is the virtue of him who made that visible [sun] fettered by any unworthiness in the minister. (8) Therefore, then, this person has only injured himself by wickedly administering the good. For the inviolable sacrament, which was given through him, held the perfection of its virtue for others.

The other reference in Denzinger looks to cross reference to 851, which is a reflects the Council of Trent

1608 851 Can. 8. If anyone shall say that by the said sacraments of the New Law, grace is not conferred from the work which has been worked [ex opere operato], but that faith alone in the divine promise suffices to obtain grace: let him be anathema.