Some theologians hold that view at least in regard to those who lived in pagan lands where the Gospel had not yet been preached.

St. Alphonsus held it as the "more probable" and "more common" opinion that belief in the Trinity is required by all by necessity of means for salvation.

However, he explains all three opinions, i.e.

1) that explicit belief in the Trinity is "necessitate medii" for all;

2) that explicit belief is "necessitate medii" for all but in some rare cases God makes exception to this;

3) that explicit belief in the Trinity is only "necessitate praecepti" and that only implicit belief in the Trinity is required "necessitate medii."

He calls this third opinion (which he lists as the second) "also probable enough" and he quotes a number of eminent theologians who hold this opinion. Here is the passage in his works:

https://books.google.com/books?id=NR48AAAAMAAJ&pg=PA296

Now they (adults) are disposed unto the said justice, when, excited and assisted by divine grace, conceiving faith by hearing, they are freely moved towards God, believing those things to be true which God has revealed and promised, and this especially, that God justifies the impious by His grace, through the redemption that is in Christ Jesus; and when, understanding themselves to be sinners, they, by turning themselves, from the fear of divine justice whereby they are profitably agitated, to consider the mercy of God, are raised unto hope, confiding that God will be propitious to them for Christ's sake; and they begin to love Him as the fountain of all justice; and are therefore moved against sins by a certain hatred and detestation, to wit, by that penitence which must be performed before baptism: lastly, when they purpose to receive baptism, to begin a new life, and to keep the commandments of God. Concerning this disposition it is written; He that cometh to God, [1] must believe that he is, and [2] is a rewarder to them that seek him; [even Trent only mentions two necessities, why didn't they check with the Church of Ladislaus first?] and, Be of good faith, son, thy sins are forgiven thee; and, The fear of the Lord driveth out sin; and, Do penance, and be baptized every one of you in the name of Jesus Christ, for the remission of your sins, and you shall receive the gift of the Holy Ghost; and, Going, therefore, teach ye all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost; finally, Prepare your hearts unto the Lord.

All of the supernatural aids granted by God to any man tend to lead him to the eternal possession of the Beatific Vision. They likewise direct him toward those realities which, either by their very nature or by God's own institution, are requisite for the attainment of the Beatific Vision. One of those realities is the visible Catholic Church, the religious society over which the Bishop of Rome presides as the Vicar of Christ on earth. The graces which God grants to any man outside the Church will inevitably guide him in the direction of the Church. Fenton

If a man continues faithful to the graces given him by God he will certainly attain to eternal salvation. And he will just as certainly obtain that salvation "within" the true Church of Jesus Christ. God's grace will lead a man in the direction of justification, according to the pattern set forth in the teaching of the Council of Trent. It will direct him to believe God's revealed message with a certain assent based on the authority of God Himself revealing. It will lead him in the direction of salutary fear and of hope and of initial love of God and of penance. Ultimately it will lead him to a desire of baptism, even though, in some cases, that desire may be only implicit in character. And baptism is of itself the gateway to the Church, the Mystical body of Christ, within which the life of grace and salvation are to be found. In the case of a man who is already baptized, the preparation for justification includes an intention (at least implicit) of remaining within the kingdom of God to which baptism itself is the gateway. Fenton

It is both idle and misleading to characterize the teaching of the Cantate Domino as in any way "rigorous" or exigent [demanding - J.G.]. This doctrine, which is standard Catholic teaching, is only the expression of what God has taught about the place of His Son's Mystical Body in the economy of man's salvation. Neither the Catholic Church itself nor the teachers of the Church have made the Church something requisite for the attainment of the Beatific Vision. When the Church makes the sort of statement that is found in the Cantate Domino, it is acting merely as the teacher of what God Himself has revealed. As the Mystical Body of Christ, the society within which Our Lord Himself is the supreme Teacher, the Church could not do otherwise. Fenton

I take it, then, LoT, that you agree with Fenton's assessment that V2 represents an improvement in the area of Catholic ecclesiology.

2. Is it required by a necessity of means or of precept to believe explicitly in the mysteries of the Holy Trinity and Incarnation after the promulgation of the gospel? 

The first opinion and more common and held as more probable teaches belief is by necessity of means; Sanch. in Dec. lib. 2. c. 2. n. 8. Valent. 2. 2. d. 1. qu. 2. p. 4. Molina 1. part. qu. 1. a. 1 d. 2. Cont. Tourn. de praeceptis Decal. cap. 1. art. 1. §. 2. concl. 1. Juven. t. 6. diss. 4. a. 3. Antoine de virt. theol. cap. 1. qu. 2. Wigandt tr. 7. ex. 2. de fide n. 22. Concina t. 1. diss. 1. de fide cap. 8. n. 7. cuм Ledesma, Serra, Prado, etc. Also Salm. tr. 21. c. 2. punct. 2. n. 15. Cuniliat. tr. 4. de 1. Dec. praec. c. 1. §. 2. et Ronc. tr. 6. c. 2. But the last three say that in rare cases it may happen that one can be justified by implicit faith only…

But the second opinion that is also sufficiently probable says by necessity of precept all must explicitly believe in the mysteries. However, for necessity of means it is sufficient to implicitly believe in the mysteries. So Dominicus Soto (in 4. sentent. t. 1. d. 5. qu. un. art. 2. concl. 2.) where he says: Even though the precept of explicit faith (in the Trinity and Incarnation) absolutely obliges the whole world, yet there also are many who are invincibly ignorant [of the mysteries] from which the obligation excuses. Franciscus Sylvius (t. 3. in 2. 2. qu. 2. art. 7. and 8. concl. 6.) writes: After the promulgation of the gospel explicit faith in the Incarnation is necessary for all for salvation by a necessity of precept, and also (that it is probable) a necessity of means…

Card. Gotti (Theol. t. 2. tr. 9. qu. 2. d. 4. §. 1. n. 2.) says: In my judgment the opinion which denies that explicit faith in Christ and in the Trinity is so necessary that no one can be justified without it is very probable. And he adds that Scotus holds this opinion…

Elbel. (t. 1. conferent. 1. n. 17.) writes today that this opinion is held by notables. DD. Castropal. part. 2. tr. 4. d. 1. p. 9. Viva in Prop. 64 damn. ab Innocent. XI. n. 10, Sporer. tr. 11. cap. 11. sect. 11. §. 4. n. 9. Laym. lib. 2. tr. 1. cap. 8. n. 5. who teach this is not less probable than the first, with Richard. Medin. Vega, Sa, and Turriano. Card. de Lugo, de fide d. 12. n. 91. calls the first speculatively probable, but defends this second view at length and in absolute terms as more probable, with Javell, Zumel, and Suarez d. 12. sect. 4. n. 10. the writings of Lugo likewise seem to be the opinion of St. Thomas 3. part. qu. 69. a. 4. ad 2. where the Doctor says: Before Baptism Cornelius and others like him receive grace and virtues through their faith in Christ and their desire for Baptism, implicit or explicit. Wherefore, argues Lugo, just as Cornelius freely obtained grace by implicit faith, so even one can obtain the same in a place where the gospel is not perfectly promulgated. He will be able in such a place to obtain the same who is invincibly ignorant of the mysteries in a place where the gospel has not been sufficiently promulgated. They say it is repugnant to the divine goodness and providence to damn invincibly ignorant adults who live uprightly in accordance with the light of nature whereas Acts 10:35 says, “But in every nation he that feareth him and worketh justice is acceptable to him.”

They respond that even though all the Scriptures and Holy Fathers’ testimonies oppose this opinion, their opinion is more easily explained by necessity of precept, or because ordinarily almost none are saved without explicit faith in the mysteries, because after the promulgation of the gospel almost no one labors out of invincible ignorance. Or that, says Lugo, they can be explained by implicit faith or explained by desire… Source: Liguori, St. Alphonsus. An Exposition and Defence of All the Points of Faith Discussed and Defined by the Sacred Council of Trent, Along With the Refutation of the Errors of the Pretended Reformers. Dublin, 1846.