You raise a several important issues. Canonicity is a very difficult area for Protestants. Roman Catholics, of course, appeal to the infallibility of church councils to affirm the Canon, but we Protestants recognize no such infallible conciliar authority. In fact, notice that the Chicago Statement does not say that the actual Canon we possess was compiled by divine inspiration. Rather, the canon "in principle" was divinely inspired. The exact meaning of this phrase is unclear, but it seems to suggest that the canonization process was not as closely superintended by God as the initial inspiration of the autographa of the books of the Bible. That is to say, the Chicago Statement seems to indicate that canonization is inspired only in the sense that all the books in it are inspired.
This raises some important issues that we need to acknowledge:
- We do not have all inspired literature in our Canon. This is evident from the "first" letter to the Corinthians (cf. 1 Cor. 5:9) and the other books cited in Kings and Chronicles (e.g. 2 Chr. 9:29). All the books we have are inspired, but we do not have all that was inspired.
- The formation of the Canon was a process. No heavenly hammer came down telling us when and where to stop. Some Old Testament books were questioned (like Esther), and some Jewish groups had other books and additional materials (e.g. extra psalms). Yet, the Canon of the Old Testament was, for the most part, settled by the time of Jesus and endorsed by him (cf. Matt. 5:17; Luke 24:27). So, the Canon of the Old Testament was basically stable for the Christian church. The New Testament was not so simple. It came to be closed in a manner much like the process of Acts 15. It took shape from a core of books known to be apostolic or closely associated with apostles (i.e. having implicit apostolic approval), primarily the Pauline epistles, and the Gospels. These were then used as a doctrinal standard to judge any other books that large groups of churches proposed to be added to the Canon. Some books like Jude and Hebrews were hotly debated, but the doctrinal issues were resolved and they were included.
- Once the early church settled on a Canon, books proposed later were never added, not because they were necessarily uninspired, but because there would have been no end to the controversies. So, the Canon became the measure of all other books (e.g. the Apocrypha), even of, hypothetically, other inspired books. This became the way for the church to secure its beliefs.
So, what about those works like the Book of Mormon that claim to be additional authoritative revelation? In my own assessment of the problems involved with establishing a Canon, I have come to somewhat different conclusions than prior thinkers. Personally, I believe that Canon ought not to be so much an issue of "true" documents as an issue of "authoritative" documents. Truth in and of itself is not authoritative; God is the source of authority.
Let me illustrate my point by referring to general revelation. General revelation is infallible and true, but it is not necessarily authoritative. Since the Fall, general revelation is not an entirely sufficient standard by which to bind our consciences because creation is not sufficiently interpretable (in my personal opinion) so as to be an absolutely sure guide (i.e. are we to emulate what we see because it reflects God's character, or are we to depart from what we see because it is the result of the Fall?). This is not to deny that general revelation is instructive and helpful, nor is it to deny that in some cases God has made certain aspects of general revelation authoritative (Rom. 1:18-2:16). Rather, it is to say that general revelation is not inherently authoritative in all its apsects.
As another example, imagine that I write the following formula on a piece of paper: 2 + 2 = 4. That formula is true, but does that mean that we ought to include that piece of paper in the Canon? No, it does not. Or suppose that I say, "It seems to me that I am hungry," and let's imagine that I am actually correct about being hungry. Does this statement about my hunger somehow become universally binding on the consciences of all men as an authority to which they must bow? Of course not.
For me, the fundamental issue behind canonicity is authority. Writings ought to be in the Canon not just because they are true, but because they also carry God's delegated authority. In the Old Testament, the men who carried God's delegated authority were the covenant administrators and certain men whom God directly called as covenant emissaries (such as the writing prophets). These men were responsible for the writings contained in the Old Testament either by writing them or by approving of their inclusion in the Canon.
In the New Testament, the Christian prophets were not covenant emissaries called directly by God and delegated with his authority. They received true revelation, but it was not authoritative. For example, in Acts 21:10 Agabus prophesied that Paul would be captured if he went to Jerusalem. Had this message been delivered by a covenant emissary, the implication would have been an authoritative instruction that Paul not travel to Jerusalem because it was against God's will. Paul, however, recognized that Agabus was not an authoritative covenant emissary, and thus felt free to take his prophecy under consideration, but not bound to do anything in particular in response to it. The apostles were the only ones who were called directly by God and who received his delegated authority. So, only the apostles were able to write or approve authoritative documents.
Similarly, I am inclined to believe that the "keys" of the kingdom were not given to the church, but explicitly to Peter and implicitly to the other apostles (Matt. 16:19). The keys belonged to God, and not to the apostles, but God delegated the power of the keys to the apostles for their use. Because the keys were not the apostles' to give, they had no authority to pass them on to anyone, and no one on earth today holds the keys (this is not the official position of Third Millennium, nor is it in agreement with traditional Reformed interpretation).
In the same way, the Old Testament covenant emissarial prophets could not choose, empower or delegate authority to other prophets. Rather, God had to choose and gift new prophets (e.g. 1 Kings 19:16). In the New Testament, only God could choose and authorize apostles, just as only God could empower men for spiritual service. Personally, I believe this is one reason that the disciples cast lots to replace Judas (Acts 1:23-26). Generally, it is argued that the disciples cast lots because they could not figure out who was the best man for the job, but this is not what the text says. Rather, it indicates that they cast lost in order that God might be the one to choose. Moreover, the disciples demonstrated in plenty of other places that they were capable of making tough descisions. Incidentally, this argument also works against apostolic succession as practiced in the Roman Catholic Church because it requires all authoritative covenant emissaries to be chosen directly by God -- even the apostles could not choose their replacements.
So, as I understand Scripture, a work is justifiably canonical only if it carries God's delegated authority. Only works written by or approved by those to whom God has delegated authority to speak on spiritual matters may be considered authoritative.
This still leaves a significant problem when it comes to the idea of extra-canonical literature. Specifically, there is no irrefutable proof in any recognized authoritative text (i.e. canonical book) that indicates that God will never call another authoritative covenant emissary, so there is no way to argue categorically against additional authoritative books. Perhaps the most common argument from Scripture is that from 1 Corinthians 13:10, where "the perfect" is taken to refer to the Canon. This argument is simply poor exegesis - that is not what "the perfect" means in the context of 1 Corinthians 13. A better argument based on Scripture is that because there can be no more apostles, there can be no more authoritative revelation. Paul's suggestion that the Corinthians especially pursue prophecy indicates that he did not expect them to pursue the even greater gift of apostleship. Paul also referred to himself as "one untimely born" (1 Cor. 15:8), indicating that his apostleship was unusual because he was called after Jesus had ascended. This strongly implies that there will be no more apostles, and thus becomes a very high hurdle for any supposed new apostles to leap. But it does not prove the point beyond all argument.
The best argument, in my opinion, is not that Scripture denies the possibility of a new authoritative covenant emissary, but the historical argument that God has not called another authoritative covenant emissary. Of course, Joseph Smith claimed that God called him directly, but that does not make his claim true. If God had called him directly -- not just by giving him true prophecy, but rather by calling him to prosecute the covenant with his people -- then we would be obligated to submit to his authority.
But because all authoritative covenant emissaries carry God's delegated authority, no subsequent emissary can contradict or invalidate the authoritative instruction/revelation of prior emissaries. In Joseph Smith's case and in the case of the Roman Catholic popes, it is abundantly evident that their words do not accord with existing authoritative revelation. This leaves only three possibilities when we assess their claims. First, they could be true emissaries and prior revelation could be false. Second, they could be false emissaries and prior revelation could be true. Third, both they and prior revelation could be false. Since we recognize contradictions between the prior revelation on the one hand, and Joseph Smith and the popes on the other hand, there is no possibility that both the self-proclaimed emissaries and the prior authoritative revelation are true. Further, our faith and reason prevent us from accepting the conclusion that the prior revelation is false. This leaves us with only one possible conclusion: the prior revelation is true, and neither Joseph Smith nor the popes were authoritative covenant emissaries.
While we cannot, in my opinion, categorically rule out all claims of new authoritative revelation, we can effectively refute all such claims that have surfaced thus far. Moreover, we have sufficient reason to believe that each book that is currently in the Canon really ought to be there.
Finally, regarding the Chicago Statement and other modern positions that assume the impossibility of new authoritative revelation, those I have encountered have been of three types: scriptural, traditional and spiritual. I have already discussed the scriptural arguments. Arguments from tradition can also be good (e.g. "the early church collected all the known apostolic works") or poor (e.g. "we haven't had any new books, therefore we can't ever have any new books"). In the category of "spiritual arguments," I would include such arguments as personal assurance by the Holy Spirit. It should be clear that while this argument may in fact be true, it is unverifiable, and thus persuasive only on a personal level. Some of these arguments offer great reasons for us to expect no further revelation. But none of these arguments can offer proof that is irrefutable in all circumstances, and thus we cannot categorically reject all claims of new revelation.
Unfortunately, this answer will not resolve the tension you feel -- I suspect it will only increase it. Nevertheless, this is not a reason for despair. It merely means that we cannot simply dismiss the claims of those who assert new revelation. Rather, we must deal with them as the Bible teaches us to deal with them, testing and evaluating the claims to see if they are true (Deut. 18:20ff.; 1 John 4;1).
This issue is a very tough one. It sounds to me like your honest investigation is causing you to reevaluate some of your assumptions. This is a good thing. It fits right in line with the idea that the Reformed faith is always reforming. Those who have gone before us have not gotten everything right, and while it may be easier simply to trust their conclusions, it may not always be right. Our tradition is not authoritative, as much as we might wish it were. Moreover, the uncertainty created by admitting the possibility of new revelation in no way challenges anything you already believe. Our Canon is admittedly a fallible collection of infallible books (we may not have included some that ought to be there), but its documents have far more historical, theological and spiritual evidence in their favor than do the spurious claims of Joseph Smith or the obvious eisegesis of the popes.
Finally, it is worth saying that Christian faith does not depend upon a book but upon a person. If we trust ultimately in documents and canons rather than in Christ, we can be shaken like autumn leaves in the wind. If in our discussions of text and canon we find our faith failing, it shows that we have placed it in the wrong place. Is this similar to the Mormon argument of the internal testimony of the burning in the bosom? Yes, a bit. That is one reason that mere men cannot convert souls -- it takes a powerful movement of the Holy Spirit to accomplish that feat.