Dr. Duguid, I always appreciate your contributions to these discussions. They certainly make us think harder about the subject.The obvious questions for this position (which is standard in EP) are "Which psalter?" and "In what other context than public worship were these psalms transmitted prior to being incorporated into the canonical psalter, given that (as you correctly state) the present psalter dates from some time after the exile?
The first psalm to be written, as you say, was that of Moses (Psalm 90). In what context was it sung prior to the formation of the psalter? Is it plausible that it was never sung in worship prior to the inspired collector adding it to the canonical book? Likewise 2 Samuel 22, which forms the basis for Psalm 18. Was the first time it was sung when it became part of an official written "hymnscroll" in the temple? Would it have been sinful to sing 2 Sam 22 or Ps 90 in worship prior to that time?
And when was that? Did David have a collection of his own psalms? The subscript to Psalm 72 suggests that there was an early collection of the Prayers of David (3-72), but that is certainly not the only pre-existing collection (the Elohistic psalter covers 42-83, bound together by a notable preference for Elohim rather than Yahweh and adding the psalms of Asaph to those of David). The subscript suggest that it was collected after David's death. There is a Korahite collection, as well as the Songs of Ascents, which cover Psalms 120-134. Books 4-5 seem likely to be a later addition, perhaps after the whole present psalter was arranged around a set of distinctly post-exilic themes (see O. Palmer Robertson's The Flow of the Psalms). What is more, there are Davidic psalms included in all the layers of the composition of the final psalter. Where were these psalms being sung, if they were not part of the then-canonical collection?? And where did the non-canonical "Psalm 151" come from, recorded in most copies of the Septuagint and in Hebrew form at Qumran?
At the very least, we have to say that "the Psalter" (as we recognize the term) did not exist in the Solomonic era, though "a psalter" very likely did. That psalter was added to and edited in a variety of ways up to an after the Babylonian exile, including "new songs" that were newly written to address the development of redemptive history and "old psalms" (including some by David) that had not been included in the original version but had carried on being transmitted (most likely by being sung in some worship context).The most obvious explanation is that "exclusive psalmody" was not practiced in the Old Testament until after the return from Babylon, at least not as currently understood. "David's prayers" probably existed as one psalter alongside several others, which may well have included other psalms that were not chosen for inclusion in the final canonical version (just as there were other Davidic psalms not included in "David's prayers").
That doesn't prove that EP is unbiblical or wrong, or that we shouldn't perhaps sing more psalms than we do, but it does highlight the fact that the argument is a whole lot more complicated than "where is Scripture do we derive the notion that we can sing man-made songs?" or "Paul was merely exhorting to the continuance of a 1300 year tradition in the Church."