My understanding of the argument is not that Paul is borrowing from the LXX, but that the LXX's usage of "hymn" in reference to psalms demonstrates that the Jews of the time did not always and only use "hymn" in reference to non-psalms. We don't even need the LXX for this. This can also be demonstrated from the New Testament.There is the claim that Paul takes from the LXX translation, and borrows the terms those translators (of the Hebrew) borrowed, and uses them as TECHNICAL TERMS in Eph/Col when he enjoins the Christian use of "Psalms, hymns, and spiritual songs." The argument goes: these verses use these three terms, but only refer to "different sorts of musical forms found in the Psalter." This argument rests on the apriori that Paul's use of these terms is a self-conscious borrowing of the language of a translation to make a theological point.
In other words, both Paul and the LXX are using the word "hymn" in a manner common to Jews, or so the argument goes. If you were walking out of synagogue on the sabbath you might have said, "I really enjoyed the hymns today," in reference to the psalms. The LXX is brought into play not because Paul is supposed to have quoted it or borrowed terms from it, but merely because it demonstrates that this was a legitimate usage of the word "hymn."