First, the freedom to pursue the question was instilled in me through my pro-inerrancy education at Westminster Seminary (though one going to WTS now would not be given the same freedom). At the time I went, there was a flourishing tradition of carefully distinguishing between the commitment to inerrancy and particular hermeneutical and/or critical conclusions.
This tradition was embodied in the title of a 1988 faculty collection, Inerrancy and Hermeneutic (ed. Harvey Conn). In that collection, Moises Silva says that even the issue of who wrote a letter, which may seem in some ways to be the most obvious conclusion to draw from an “inerrant” Bible (I mean, if you can’t believe the “From” line of the letter, what can you believe?), is an issue that must be decided based on historical evidence. And, if a letter is found to be pseudepigraphical, then our understanding of what it means to affirm an inerrant Bible must be shaped so as to allow for that.
This is reflective of an important factor that drives a lot of my work: that no theology worth holding is going to so exert its control over our reading of the Bible that it will forbid us from saying what good exegesis of the passage demands that we say.
In this case, I don’t find persuasive that there is much theologically at stake for recognizing that Paul did not write these letters. They are scripture and therefore we have them as part of the canon, the rule of the church’s faith and life.
For me, the most compelling arguments in favor of Pauline authorship are those compiled by Luke Timothy Johnson in his The Writings of the New Testament (which I have summarized elsewhere).
In my opinion, the most compelling methodology for deciding whether any New Testament letter is in fact pseudonymous is that proposed by Richard Bauckham in “Pseudo-Apostolic Letters” (JBL 107/3  469–94). Briefly, Bauckham observes that, if the stated author is in some sense a fiction, then likewise the implied audience is equally fictitious. Therefore, a genuinely pseudonymous letter will use certain literary strategies to bridge the gap between the proposed audience within the letter (i.e., contemporaries of one of the apostles) and the actual audience living one or more generations later. This may be as simple as staying on a general level (and thus the more securely we find we can speak about the particularities of the church situation being addressed, the more likely we’re dealing with a genuine letter), historical scene-setting so readers can discern the connections between the supposed author’s time and their own (see 2 Tim 2:17-18), direct commands to pass the apostolic teaching on to future generations (see 2 Tim 2:2), and/or casting certain teachings as predictions of things to come (see 1 Tim 4:1; 2 Pet 2:1-3).
Bauckham’s conclusions (491) broadly confirm mainstream scholarly assumptions:
- 2 Peter “can be regarded with very high probability as pseudepigraphal”
- The Pastorals “could very well be pseudepigraphal”
- Ephesians, James, 2 Thessalonians: “our criteria make pseudepigraphy a possibility, without deciding the issue.
- All other NT letters: “the question of authenticity remains in these cases an open one which must still be decided on other grounds. But it seems that any other case for pseudepigraphy among the NT letters is by these criteria very implausible indeed.
For me, the pseudonymity of the Pastorals is an open issue but not a terribly important one. As Daniel says, they’re in the canon either way and therefore count as inspired Scripture. What I usually tell my class is that on Mondays, Wednesdays, and Fridays I think Paul wrote them; on Tuesdays, Thursdays, and Saturdays I think they’re pseudonymous; and on Sundays I pray because I’m not sure what I think!