For Christians who are skeptical of the writings of the Apostolic & Ante-Nicene Fathers, what is their epistemological basis for trusting a canon?

Simply because the canon, when corruptions are recognised, removed and the text restored to the originally inspired intention of what God inspired, we see a narrative without contradiction and wholly self-interpretive.

Trusting in God’s provided word/canon is made substantially easier when compared to the various constructs derived from men alone who, obviously not inspired by God, have provided material clearly incongruent to the original text.

“broad agreement among them”

This supposed broad agreement took 100’s of years to obtain. God has no reason to develop truth, He simple reveals it and has no need to continually refine it.

Having an inspired work/canon that encompasses millennia and foretells a future that unfolds perfectly is not to be compared with a humanly derived construct without any credentials at all!

Anyone can write a mysterious story about a complicated god who is derived from pagan mythology, but it takes the true God to predict the future and bring it about without deviation. Thus we can reliably trust in the future yet to be realised when the same true God is writing the only true story.

Trusting in the canon over other writings is, by those reasons, a no-brainier.

Epistemology is the study of knowledge and rationality of belief. What is the basis for trusting a canon, if you can't rely on outside writings?

The bible provides its own requirement to be considered as God authored and thus for trusting its contents. For the New Testament, all books had to be written during the times of the apostles by either the eyewitness or associate. For the Old Testament, the canon had to be written during the time of a valid prophetic line.

The Muratorian Fragment mentions these internal requirements.

it [Shepard of Hermas] cannot be read publicly to the people in church either among (79) the Prophets, whose number is complete, [8] or among (80) the Apostles, for it is after [their] time. -source-

From where did its author circa 170 CE know of these requirements?

4. From the time of Artaxerxes to our own day all the events have been recorded, but the accounts are not worthy of the same confidence that we repose in those which preceded them, because there has not been during this time an exact succession of prophets.679 679 The Artaxerxes here referred to is Artaxerxes Longimanus who reigned b.c. 464 to 425. It was under him that Ezra and Nehemiah carried on their work and that the later prophets flourished. Malachi—the last of them—uttered his prophecies at the end of Artaxerxes’ or at the beginning of Darius’ reign. It was commonly held among the Jews that with Haggai, Zachariah and Malachi the prophetical spirit had departed from Israel, and the line was sharply drawn, as here by Josephus, between them and the writers of the Apocrypha who followed them. -source-

Josephus was a contemporary of Christ. It was already known. What did Christ tell us?

The Hebrew Bible was numbered from Genesis to Chronicles and here. So, when Jesus outlined the canon from Abel to Zacharias, He referenced the canon of the time.

unto the blood of Zacharias son of Barachias, whom ye slew between the temple and the altar--As there is no record of any fresh murder answering to this description, probably the allusion is not to any recent murder, but to 2Ch 24:20-22, as the last recorded and most suitable case for illustration. And as Zacharias' last words were, "The Lord require it," so they are here warned that of that generation it should be required. -source-

As well, Jesus defined the Law, Prophets, Writings with the same definition of the Hebrew Canon.

The New Testament shows that its canon was none other than that which exists to-day. None of the Apocrypha or Pseudepigrapha is ever quoted by name, while Daniel is expressly cited in Matt. xxiv. 15. Matt. xiii. 35 (= Luke xi. 51) proves that Chronicles was the last canonical book. The statement, "That upon you may come all the righteous blood shed upon the earth, from the blood of righteous Abel unto the blood of Zacharias," contains a reference to II Chron. xxiv. 20. The three chief divisions are enumerated in Luke xxiv. 44—"Law," "Prophets," and "Psalms"—as they are in Philo. -source-

Paul mentions that to the Jews were entrusted the words of God (Rom 3:2).

And of course there is prophecy and fulfilled prophecy.

I'll provide some proof later about the New Testament.

"Despite the obvious high levels of human intervention that were required"

I'm not sure I agree with this. If you really have Gospel accounts from the Apostle Matthew, the Apostle John, the Apostle Paul (at a step), and the Apostle Peter (at a step), it is probably a pretty straightforward argument that these should be kept. Who should you listen to if not the Apostles or their close associates?

But the whole point is that it's not a work of human hands, but of divine composition, as @Rajesh says. It's not clear to me how this question doesn't collapse into the general question of "How do you know the NT is divinely inspired?" That's a question for everybody, whether they trust various later but still early writers or not.

Having said that, if one reads many of these early writers pretty much any contemporary Christian will find much to disagree with. I think a better question would be: who isn't skeptical of various theories and beliefs amongst those early writers? To take one of your examples, 'the deity of Christ', this just seems wrong. The picture of early writers (2nd and 3rd centuries) seems much more complex, where almost all of them would be tossed out of contemporary Trinitarian congregations for one heresy or another related to their theologies or Christologies!

However, you can push on authorship. If you don't trust the early writers, how do you know the Apostles and close associates wrote the Gospels? Isn't that based on testimony of the early writers?

First, the early writers very well may be wrong on all sorts of things, including history. (I'm looking at you, Eusebius!) There are 3 important points here, and I'll illustrate the points with a more contemporary example below.

1. Accepting the canon doesn't require depending on specific early writers. It depends on tradition in a broader sense.

2. Accepting specific traditional claims about, say, authorship doesn't mean you accept all the theology or Christology of those writers. Nor does it mean those authors support all of or even the bulk of the evidential basis for that belief.

3. I would want to hear a good reason for questioning the traditions of attributing authorship - they could be wrong but I haven't heard particularly compelling reasons yet. These are not abstruse theologies that are seemingly being innovated by particular writers, but central, basic traditions.

I want to illustrate this with an example. Let's say you have a contemporary Lord of the Rings Society. We can call people who belong to this Lotrs.

If you asked a Lotr who wrote the Lord of the Rings, almost anyone could tell you it's J.R.R. Tolkien. This is a basic, central fact for Lotrs.

Now let's say you asked a random Lotr which year Return of the King was published. The answers you get might vary.

Now let's say you asked a random Lotr about who, exactly, Gandalf is. They would almost all agree on certain aspects (he's a wizard, he can use magic, he's a leader of the good guys, and so on). Yet, some might have a 'high' Gandalfology, holding Gandalf is in some sense Iluvatar, others that Gandalf is a 'god', and others that Gandalf is an angelic being.

Now, let's imagine we're 1,000 years in the future. Almost all of what Lotrs wrote in the distant past has been lost. The future scholars are trying to figure out the basis of their belief that J.R.R. Tolkien wrote the Lord of the Rings. The earliest writing they have mentioning this is by an early Lotr named Sir Randolph Imhurst (SRI), for whom a few letters have survived.

It would be questionable to think the reason their tradition of J.R.R. Tolkien being the author has been received as it is, is because of SRI's ancient writing. Rather, SRI's writing is symptomatic of the cause, which is a common belief held by many in ancient society, and which was passed through the generations down to our future Lotrs.

But let's say SRI said in one of his letters that Return of the King was published in 1965. Would it be reasonable to be less certain of that claim than SRI's claim that J.R.R. Tolkien wrote Lord of the Rings? It seems obviously 'yes', people are more likely to get dates like that wrong.

And let's go further and say that SRI also wrote, in these preserved letters, about how Gandalf was actually Iluvatar, not just an elevated spiritual being. Yet, in another of our fragments by another ancient Lotr, we read that Gandalf was a subsidiary being, different from Iluvatar, more appropriately called a 'god'. And in yet another that Gandalf is actually an angel.

Would having doubts about SRI's theological speculation re Gandalf mean we also have to doubt his claims about authorship?

It seems obvious the answer is 'no'.

The folowing extract from the Westminster Confession (1.5) may be of interest in regard to this question. The Westminster Confession is the Doctrinal Standard in many Reformed churches. These are accepted in most Presbyterian denominations in Scotland including the national Kirk itself, and many other churches worldwide.

Although identifying several reasons to believe the Holy Scriptures including the tradition of the Church, the consent (agreement) of all its parts and many other excellencies, it goes on to say that these alone are not, in themselves, fully persuasive. Rather, full persuasion comes from the work of the Holy Spirit in the heart of the believer.

The authority of the holy Scripture, for which it ought to be believed and obeyed, dependeth not upon the testimony of any man or church, but wholly upon God (who is truth itself), the Author thereof; and therefore it is to be received, because it is the Word of God. We may be moved and induced by the testimony of the Church to an high and reverent esteem of the holy Scripture; and the heavenliness of the matter, the efficacy of the doctrine, the majesty of the style, the consent of all the parts, the scope of the whole (which is to give all glory to God), the full discovery it makes of the only way of man’s salvation, the many other incomparable excellencies, and the entire perfection thereof, are arguments whereby it doth abundantly evidence itself to be the Word of God; yet, notwithstanding, our full persuasion and assurance of the infallible truth and divine authority thereof, is from the inward work of the Holy Spirit, bearing witness by and with the Word in our hearts.