How does the Westminster Confession address the paradox of the Bible canon?

Frankly, but only in my opinion, I think you may be over-complicating things a tad.

To say there is an inherent paradox in Chapter I, Article IV of the Westminster Confession (hereafter, WC) is simply not accurate.

In writings generally and not just in religious writings, for there to be a paradox there needs to be either a real contradiction or an apparent contradiction. What is contained in the Westminster Confession and other universally agreed upon statements of faith (e.g., The Apostles' Creed) is not a paradox but a defense of a presupposition; namely, the Bible is God's Word, and its authority derives ultimately from God.

What has been given short shrift in the WC is the role and instrumentality of the writers of Scripture, who because they were, in large part, prophets (in the Tanakh) and apostles (in the New Testament), the books which found their way into the canon of Scripture did so because they were recognized by literate Jews and Christians to be the Word of God, and not just the word of men.

Peter realized this fact when he said,

So we have the prophetic word made more sure, to which you do well to pay attention as to a lamp shining in a dark place, until the day dawns and the morning star arises in your hearts. But know this first of all, that no prophecy of Scripture is a matter of one’s own interpretation, for no prophecy was ever made by an act of human will, but men moved by the Holy Spirit spoke from God (2 Peter 1:19-21 NASB, my emphasis).

Additionally, Peter recognized the apostolic authority which resided in the apostles' teachings in general and in the words and teachings of the apostle Paul in particular because those teachings came from the Master directly and sometimes indirectly. As for the Master's deeds, the disciples and apostles were eyewitnesses (see, for example, Luke 1:2 and 2 Peter 1:16). First, the prophets and apostles in general:

This is now, beloved, the second letter I am writing to you in which I am stirring up your sincere mind by way of reminder, that you should remember the words spoken beforehand by the holy prophets and the commandment of the Lord and Savior spoken by your apostles (2 Peter 3:1-2 NASB, my emphasis),

We also have the prophetic message as something completely reliable, and you will do well to pay attention to it, as to a light shining in a dark place, until the day dawns and the morning star rises in your hearts. Above all, you must understand that no prophecy of Scripture came about by the prophet’s own interpretation of things. For prophecy never had its origin in the human will, but prophets, though human, spoke from God as they were carried along by the Holy Spirit (2 Peter 1:19-21 NIV, myemphasis).

Interestingly, Peter considered the "prophetic message" as contained in the Tanakh's Law, Prophets, and Writings to be on a par or even superior to his own eyewitness testimony! Peter, in my opinion, was not just being modest in his apparent discounting of his testimony as an eyewitness but was simply reflecting the confidence which Jews at the time had in the Tanakh.

Now, for Paul's apostleship in particular:

Therefore, beloved, since you look for these things, be diligent to be found by Him in peace, spotless and blameless, and regard the patience of our Lord as salvation; just as also our beloved brother Paul, according to the wisdom given him, wrote to you, as also in all his letters, speaking in them of these things, in which are some things hard to understand, which the untaught and unstable distort, as they do also the rest of the Scriptures, to their own destruction (2 Peter 3:14-16 NASB, my emphasis).

Peter may not have known that his two letters would one day find themselves in the canon of Scripture, but he certainly did know with full conviction and assurance that the apostle Paul's teaching had the apostolic ring of truth to it. Peter put Paul's teaching on the same level as "the rest of Scriptures." (A good question to ask at this juncture would be, "To Peter, what constituted 'the rest of Scriptures'?" but let's not complicate things any more than they need to be!)

In short, though the canon of Scripture came to fruition over the course of centuries, what constituted its various parts as being God's Word was the acknowledgment of the church universal (beginning in Jerusalem, of course) that some writings were "Scripture" simply because they originated in the apostles' teaching.

We must not forget at this point, however, that God was at work through his Holy Spirit in the early days of the church in the dual processes of the inspiration of his Word and the preservation of his Word. In a sense, what God did by first providing and preserving his Word in the Tanakh, he simply repeated by inspiring and preserving his Word in the New Testament writings.

The process by which the canon of Scripture was codified, preserved, and transmitted faithfully over the centuries, a process which included the creation of creedal statements such as the WC and The Apostles' Creed, may seem to smack of circular reasoning at best and paradox at worst, but I believe it does not. Positing a belief (and the statements which support that belief) on bedrock assumptions or presuppositions, as the writers of the WC did in Article I, Chapter IV, did not involve a statement that is self-contradictory or logically untenable, which is at the heart of a true paradox. Rather, the positing of that belief was based on a valid deduction from acceptable premises.

By and large, the writers of Scripture claimed either to be speaking and writing on God's behest ("Thus saith the LORD," the original prophets said repeatedly) or they claimed to be repeating the words, teachings, and deeds of Jesus faithfully and with the corroboration of all the apostles. Once the apostles, disciples, and other eyewitnesses of the life and teachings of Jesus had died, God was obviously at work in preserving his church by preserving his Word and the faithful transmission of it through each successive generation of Christians.

False teachers arose in the first century AD within years of Christ's ascension, but faithful teachers, preachers, translators, copyists, church councils (and much later, printers!) then made sure that subsequent generations of Christians would have a faithfully transmitted copy of God's precious and eternal Word. This process does not involve paradox or circular reasoning; it's just the way God decided to reveal his Word to the world.

The Westminster Confession of Faith and its associated catechisms (collectively referred to as the Westminster Standards) are designed to teach the Reformed faith, but not necessarily to defend it. Even the inclusion of the footnoted scripture proofs was at the request of the English House of Commons and would not have been included otherwise (it was adopted without amendment by Scotish General Assembly).

The Westminster Standards were designed to be teaching tools and governing documents for the church bodies who adopted them to the point that through the modern era they are binding in the church constitutions. Rather than trying to anticipate and respond to all conceivable objections, they wanted to summarize the theological system of the Reformed faith itself and leave its defense to ministers, evangelists, and teachers. As decently and orderly ministers, they kept full minutes in the assembly which are still available today. This means that if you have the will for it, you can examine the reasoning behind why the documents were written the way that they were.

As an integral part of the church constitution, it actually makes for a more powerful document to keep the beliefs separated from their defense. It also makes the didactic aspect more flexible, since in different times and places learners will have different hangups, and the inclusion of specific defensive arguments would likely seem antiquated, foreign, or irrelevant. It leaves room for the teacher to teach.

As for the alleged paradox of the canon, it only really shows up in the context of an exaggerated form of sola scriptura that divorces scripture from the Author. In the Reformed system of doctrine, scripture derives its authority from the divine author. The question of how Christians know which books are scripture has nothing to do with scripture's authority. Scripture is ultimate only because it proceeds from the Ultimate.

There are other divine revelations that are not in the Biblical canon, and are still the word of God. A few examples of non-canonical words of God are the prophetic utterances in the congregational worship of the New Testament church, the prophetic office during the Israelite theocracy when the words were not recorded, some of the Mosaic case law, and the primordial revelation that predated Moses but allowed people like Seth, Enoch, Noah, and Shem to live in fellowship with God. All of this material is "lost" to us, according to the plan of God, because its significance was not for us. Like the canon itself, these revelations did not depend upon the canon to affirm them. The only reason they are not a part of the Reformed system of doctrine that we don't have them.

Scripture does not require an inspired table of contents in order to have a foundation for knowing the books because scripture's authority is not self-referential.

As others have mentioned, there are reformed argument for the Protestant canon that will address these topics in greater detail. Adrian Keister suggested the Richard Gaffin series, and I would add Michael J. Kruger, the current president of Reformed Theological Seminary Charlotte, whose research is centered on this question and he also has a lay level book titled "Canon Revisted."

As Adrian Keister also mentioned, WCF:I.5 demonstrates the Westminsterian distinction between what makes scripture of divine origin (God speaking through human authors) and how a person comes to be convinced of it. Notice the importance of this distinction. As finite creatures, we can only become convinced of things. We cannot know them directly as God does. God knows what the true canon is and he expects us to be able to reliably apply that knowledge to read scripture devotionally. This gives us all the more reasons to examine the arguments for the canon. It also justifies our use of the Bible in testing our view of the canon without being circular, since our apprehension of the correct canon does not determine the actual truthfulness of the canon. Though it may sound obtuse to use the canon to confirm the canon, it is eminently practical. You can instantly eliminate the gnostics writings since they are incompatible with the gospels, Paul, and the Old Testament.

One basic argument (Michael J Kruger's) is that you can examine the books by criteria: divine qualities, apostolic origin, and community reception. Again, this argument is not itself present in the Westminster Standards directly, but it should be valuable to learn how experts on the Westminster Standards and the canon have addressed the question. Of course, there is more detail in the books.

Yes, the Westminster Confession of Faith answers this apparent, but not real, circularity as follows. In Chapter 1 Paragraph 5, we read:

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 does 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.

The last clause (starting with "yet notwithstanding, our full persuasion...") is the key. The Holy Spirit convinces the believer that the Bible is true. This is the epistemology of the Christian. It is not circular, because it has an infinite reference point starting with God. God has to reveal himself before anyone knows anything about him.

Incidentally, the Bible actually does talk about itself, particularly in Luke 24, 2 Timothy 3:16-17, and 2 Peter 3:16. If you mean that the Bible does not define its own Table of Contents exhaustively, I would agree with you. The early church recognized the canon, it did not define the canon. This is consistent with John 10:27. There is a good lecture series by Richard Gaffin on how the canon originated.