What was St. Robert Bellarmine teaching on human cooperation with grace? What did St. Robert teach on grace and cooperation with it? Did he accept the Thomistic doctrine of grace? Did he agree with Báñez or did he side with Molina?
St. Robert's opinion
St. Robert's views are found in his De gratia et libero arbitrio libri sex.
According to Pohle-Preuss's Grace, Actual and Habitual: A Dogmatic Treatise (Chapter III. Grace In Its Relation To Free-Will, Section 2. Theological Systems Devised To Harmonize The Dogmas Of Grace And Free-Will, Article 2. Molinism And Congruism),

Cardinal Bellarmine, who was a champion and protector of P. Molina, seems to have rejected Molinism765 in favor of Congruism.766


Cfr. his treatise De Gratia et Libero Arbitrio, I, 12 […]: “Prima opinio eorum est, qui gratiam efficacem constituunt in assensu et cooperatione humana, ita ut ab eventu dicatur gratia efficax, quia videlicet sortitur effectum et ideo sortitur effectum, quia voluntas humana cooperatur. Itaque existimant hi autores, in potestate hominis esse ut gratiam faciat esse efficacem, quae alioquin ex se non esset nisi sufficiens.”Their first opinion is that efficacious grace consists in human assent and cooperation, so that from its occurence it is called efficacious grace, because it works its effect and thus achieves the effect, because the human will cooperates. So these authors think that it is within human power that grace is made efficacious, which otherwise of itself is only sufficient.Bellarmine treats this opinion as the extreme counterpart of Thomism (which he also combats) and disposes of it thus: “Haec opinio aliena est omnino a sententia b. Augustini et, quantum ego existimo, a sententia etiam Scripturarum divinarum.” (l.c.)This opinion is entirely foreign to the thought of St. Augustine and, as far as I think, even to the thought of Holy Scriptures. Among the Scriptural texts which he quotes in support of this view are John VI, 45, 1 Cor. IV, 7, Rom. IX, 11.

The learned Cardinal describes the difference between Congruism and extreme Molinism (which latter, it may be remarked, was not defended by Molina himself) as follows: “Neque enim intelligi potest, quo pacto gratia efficax consistat in illa interna suasione, quae per liberum arbitrium respui potest, et tamen infallibilem effectum habeat, nisi addamus, Deum iis quos efficaciter et infallibiliter trahere decrevit, eam suasionem adhibere quam videt congruere ingenio eorum et quam certo novit ab eis non contemnendam.” (Op. cit., […])Nor can it be understood that efficacious grace consists in that internal persuasion which by freewill can be rejected and yet have an infallible effect, unless we add that God efficaciously and infallibly decreed they draw it, which persuasion He sees congruent with their innate nature and which He certainly knows they will not despise.The objection that this explanation eventually resolves itself into the Molinistic theory which he had censured, Bellarmine meets as follows: “Respondeo sententiam nostram, quam S. Augustini esse demonstravimus, aliqua in re cum prima illa opinione convenire, sed in multis ab illa discrepare. Convenit enim in eo quod utraque sententia gratiam sufficentem et efficacem ponit in auxilio excitante potissimum, non in adiuvante. Sed discrepant inter se, quod prima opinio vult efficaciam gratiae pendere a voluntate humana, nostra vero pendere vult a voluntate Dei.” (l.c., cap. 13 [p. 294].)I answer that our opinion, which we have demonstrated is St. Augustine's, really agrees with that first opinion, but in many ways disagrees with it. It agrees with both opinions by putting sufficient and efficacious grace in the most powerful eliciting divine help, not in the one helped. But they disagree with each other, the first opinion wanting efficacious grace to depend on the human will, but ours wanting it to depend on the will of God.



(Improvements of the English translations, which are not in Pohle-Preuss, are welcome. ☺)
From this answer: St. Augustine's view on efficacious grace, from his De praedestinatione sanctorum, chap. 8:

Grace which is not rejected by any hard-heartedness, since it is bestowed, in the first place, to remove hardness of heart.

and his De gratia Christi, chap. 24:

[Efficacious grace is the] internal, hidden, wonderful, and ineffable power by which God effects in the hearts of men not only true revelations but even upright wills.

Congruism
Fr. Hardon, S.J., defines &quot;congruism&quot; as:

The theory of man's co-operation with grace, first developed by Francisco Suárez (1548-1617) and St. Robert Bellarmine (1542-1621) and later adopted by the Jesuit order. According to congruism, the difference between efficacious and sufficient grace lies not only in the consent of the free will (Molinism), but also in the congruity or suitableness of a particular grace to the peculiar conditions of the one who receives the grace. When the grace suits the interior dispositions and external circumstances of a person, it becomes effective by the free consent of the will; otherwise it remains ineffective because it lacks free acceptance. As in Molinism, God foresees the congruity of the grace and its infallible success. Unlike Molinism, congruism places the emphasis not on man's freedom but on the supremacy of the divine will in determining salvation. (Etym. Latin congruitas, congruity, fitness, suitability, becomingness.)