What positions with regard to epistemology exist in Catholicism?

As I understand your question, it deals with the identification of the medium of divine Revelation. I will divide my answer in two parts, providing some broader context in the first.

Stages and structure of the medium of divine Revelation

In the case of "sealed" views of divine Revelation, such as those held by Jews, Christians and Muslims, i.e. of a divine Revelation that has been fully completed in the past, the medium of Revelation which must be identified includes two stages:

• the original medium through which God has revealed in the past, and

• the proximate medium that currently holds the "deposit" of what God has revealed through the original medium and, in some views, provides authoritative identification and interpretation of that Revelation.

Regarding the original medium, and focusing on the "ultimate" such medium, Jews, Christians and Muslims identify it with Moses, Jesus Christ and his Apostles, and Mohammed respectively.

Regarding the proximate medium, there are two main views on its structure:

A. It is a book plus a tradition, both interpreted by a divinely assisted and authoritative magisterium, which is the belief held by Rabbinic Jews and Roman Catholics/Eastern Orthodox.

B. It is just a book, which is the belief held by Karaite Jews, Christian Protestants and Quranist Muslims.

The views of the main denominations of Islam, i.e. Sunni and Twelver Shia, seem to be intermediate between these two, as they recognize the Quran and also the oral traditions about Mohammed (hadith), which were eventually written down in collections of books, different for each denomination.

In the Catholic Church, the doctrine of the proximate medium of Revelation was stated magisterially first by the Ecumenical Council of Trent and then by the Ecumenical Council Vatican I, in its Dogmatic Constitution "Dei Filius", which quotes Trent:

"Furthermore, this supernatural revelation, according to the faith of the universal Church, declared by the holy Synod of Trent, is contained "in the written books and the unwritten traditions which, received by the Apostles from the mouth of Christ himself, or from the Apostles themselves by the dictation of the Holy Spirit, have come down to us, transmitted as it were from hand to hand" [Council of Trent]." (ch. 2 "On Revelation")

"For the doctrine of faith, which God has revealed, has not been proposed as a philosophical invention to be perfected by human ingenuity, but has been delivered as a divine deposit to the Spouse of Christ, to be faithfully guarded and infallibly declared (fideliter custodienda et infallibiliter declaranda)." (ch. 4 "On faith and reason")

Identification of the medium of divine Revelation

Clearly the "strict" definition of faith, e.g. by the Ecumenical Council Vatican I in its Dogmatic Constitution "Dei Filius", ch. 3 "On faith":

"The Catholic Church professes that this faith, which is the beginning of human salvation, is a supernatural virtue, by means of which, with the inspiration and assistance of the grace of God, we believe that the things revealed by Him are true, not because the intrinsic truth of the things has been perceived by the natural light of reason, but because of the authority of God Himself who reveals them, who can neither deceive nor be deceived."

cannot apply to the identification of the medium of divine Revelation, lest the epistemic situation be circular, as a person should have to identify M as the medium through which God reveals by an assent to the truth that "God reveals through M" based on the authority of God who revealed (through M) said truth (i.e. that He reveals through M)! Thus, the medium of divine Revelation must be rationally identified, based on its motives of credibility.

"Dei Filius", in line with its quoted teaching on the proximate medium of divine Revelation, states clearly that only the Catholic Church exhibits real motives of credibility of being such a medium:

"And, that we may be able to satisfy the obligation of embracing the true faith and of constantly persevering in it, God, through his only-begotten Son, has instituted the Church, and has bestowed on it manifest notes of that institution, so that it may be recognized by all as the guardian and teacher (custos et magistra) of the revealed word. For to the Catholic Church alone belong all those things, so many and so marvelous, which have been divinely established for the evident credibility of the Christian faith. Moreover, the Church by itself, because of its marvelous propagation, its exceptional holiness, and its inexhaustible fruitfulness in all that is good, because of its Catholic unity and invincible stability, is a great and perpetual motive of credibility and an irrefutable evidence (testimonium) of its own divine mission." (ch. 3 "On faith")

http://www.ccel.org/ccel/schaff/creeds2.v.ii.i.html

Addendum: A podcast of a lecture on the motives of credibility mentioned in the last quote from Constitution Dei Filius by Dr. Lawrence Feingold, Associate Professor of Philosophy & Theology at Kenrick-Glennon Seminary in Saint Louis, Missouri, is available here:

http://www.calledtocommunion.com/2013/11/lawrence-feingold-the-motives-of-credibility-for-faith/

The handout provided at the lecture is available as a pdf file here:

http://www.hebrewcatholic.net/wp-content/uploads/2013/11/12.07-Motives-of-Credibility-for-Faith-pdf.pdf

Any epistemology that respects reason and its ability to reach actual Truth, is acceptable. The original question seems based on some misconceptions. One is that the knowledge of God is not just from Faith, it is from reason, as it was for so many pagans and Greek philosophers. Cicero affirms that very few of the classical thinkers were agnostic or atheist. Aquinas is clear on this: there are certain premises that are known either in themselves or can be known by philosophic investigation. These premises can be known by the pagan. Saint Thomas Aquinas explains:

The existence of God and other like truths about God, which can be known by natural reason, are not articles of faith, but are preambles to the articles; for faith presupposes natural knowledge, even as grace presupposes nature, and perfection supposes something that can be perfected. Nevertheless, there is nothing to prevent a man, who cannot grasp a proof, accepting, as a matter of faith, something which in itself is capable of being scientifically known and demonstrated.

Summa Theologiae Ia, q. 2, a. 2, ad 1.

Saint Thomas Aquinas calls our natural knowledge of things the “preambles of faith” or the “presuppositions of faith” (cf. III Sent. d. 24, a. 3, sol. 1). The preambles or presuppositions of faith include the premises that God exists, that God is one, incorporeal, and intelligent.

But most important is the idea of LOGOS, which underlies all of Western Civilization, religious and secular -- a faith that underlies faith and reason, if I can put it that way

"the West is ultimately about reasoned inquiry in search of truth, and its way of searching for truth has been profoundly shaped by the manner in which Judaism and then Christianity affirmed the idea that God of the Bible is a rational being—the Logos. That understanding of God made all the difference, not least because it rescued the Greek and Roman world from the nonsense of pagan religion and fit well with emerging Hellenic insights into disciplines like mathematics and the natural sciences."

AND THIS OPPOSES EVERYTHING ELSE

if you have a voluntarist view of God and the world—that God’s essence is pure Will (voluntas)—then a concern for the reasonability of God, and therefore for human rationality, becomes unimportant if not irrelevant.

Source: Dr. Samuel Gregg, in a Catholic World Report 2019 article "Without Logos, the West is over." A conversation with Dr. Samuel Gregg.