As OP points out, the distinction here is between a predicate adjective (all scripture is God-breathed) and an attributive adjective (all God-breathed scripture...).1 This is well-trodden territory in the commentaries. I will summarize several arguments in favor of the latter, which mostly come from older literature.2
Bernard understood θεόπνευστος as parallel to ἱερὰ γράμματα ("holy writings") in the previous verse. Because "holy" in v. 15 is clearly attributive, he believed that "God-breathed" in v. 16 would be most naturally understood as attributive also (i.e. "God-breathed scripture). Further, he opined that for Paul to make a direct affirmation that scripture is “God-breathed,” would be irrelevant at this point in his argument, which is focused on the functional significance rather than its origin or ontology.
Others (e.g. Roberts) have appealed to the use of
πᾶς + noun + adjective elsewhere in the NT, noting that in all such cases the adjective is attributive.
There are twenty-one instances in the New Testament in which pas is used to modify a noun which is immediately followed by another adjective as in 2 Tim. 3:16. In every case the Greek order of words is (1) pas, (2) the noun, and (3) the adjective. Typical examples are "every good tree" (Matt. 7:17); "every idle word" (Matt. 12:36); "every spiritual blessing" (Eph. 1:3); "every good gift" (James 1:17). . . . In no case of this usage is the adjective separated from the noun so as to be taken as a predicate.
For a summary of arguments in favor of the attributive understanding as well as detailed refutation on the basis of syntactical considerations (which, it is argued, contra many commentators, are decisive), see Dan Wallace's The Relation of θεόπνευστος to γραφή in 2 Timothy 3:16.
To this discussion, we should note the caveat expressed incisively by Luke Timothy Johnson:
The debate is not central to Paul’s point, which concerns the function rather than the origin of Scripture, and is generated by theological concerns about the inspiration of Scripture that are driven by an anachronistic literalism.
1. Although the comments bring up the choice of rendering "inspired of God" vs "God-breathed", I don't believe any meaningful distinction was intended by the translators (though admittedly "inspired of God" uniquely allows for the word order chosen by the ASV, due to quirky English rules about unmarked relative clauses), so I have not focused on that.
2. My impression is that the modern literature generally understands the adjective as predicative but opines that the syntax is ambiguous. Wallace is an exception.
Bernard, J. H., The Pastoral Epistles. CGTSC; Cambridge, 1899; reprinted Grand Rapids, 1980 (Thornapple Commentaries), p 137.
Luke Timothy Johnson, The First and Second Letters to Timothy, The Anchor Yale Bible; (YUP, 2001), 423-424.
J.W. Roberts, Every Scripture Inspired of God, Restoration Quarterly 5 (1961), 33–37.
Daniel B. Wallace, The Relation of θεόπνευστος to γραφή in 2 Timothy 3:16. https://bible.org/article/relation-2-timothy-316#_ftnref1 Accessed 4/25/2018.