The question is based on the unverified assumption that the Greek words, κιθαρις (cithara), ψαλτηριον (psaltery) and συμφωνια (symphonia), are the source of the Aramaic and thus loan words. This is possible inasmuch as the Greek language is older than the 6th century BC and was highly influential by then. Given that the meeting of officials described in Dan 3 was truly international, and thus some international words were used.
However, It is entirely possible, in fact probable, that Greek borrowed the words from Aramaic because Aramaic appears to be much older than Greek.
Barnes makes similar comments on Dan 3:5 -
It has been alleged that this word is of Greek origin, and hence, an
objection has been urged against the genuineness of the book of Daniel
on the presumption that, at the early period when this book is
supposed to have been written, Greek musical instruments had not been
introduced into Chaldea. For a general reply to this, see the
introduction, section I, II, (d). It may be remarked further, in
regard to this objection,
(1) that it is not absolutely certain that the word is derived from
the Greek. See Pareau, 1. c. p. 424, as quoted in Hengstenberg,
"Authentic des Daniel," p. 16.
(2) It cannot be demonstrated that there were no Greeks in the regions
of Chaldea as early as this. Indeed, it is more than probable that
there were. See Hengstenberg, p. 16, following.
The Keil and Delitzsch Biblical Commentary on the Old Testament has more information -
The great delight of the Babylonians in music and stringed instruments
appears from Isaiah 14:11 and Psalm 137:3, and is confirmed by the
testimony of Herod. i. 191, and Curtius, Daniel 3:3. קרנא, horn, is
the far-sounding tuba of the ancients, the קרן or שׁופר of the Hebr.;
see under Joshua 6:5. משׁרוקיתא, from שׁרק, to hiss, to whistle, is
the reed-flute, translated by the lxx and Theodot. σύριγξ, the
shepherd's or Pan's pipes, which consisted of several reeds of
different thicknesses and of different lengths bound together, and,
according to a Greek tradition (Pollux, iv. 9, 15), was invented by
two Medes. קיתתס (according to the Kethiv; but the Keri and the Targ.
and Rabbin. give the form קתרס) is the Greek κιθάρα or κίθαρις,
harp, for the Greek ending ις becomes ος in the Aramaic, as in many
similar cases; cf. Ges. Thes. p. 1215. סבּכא, corresponding to the
Greek σαμβύκη, but a Syrian invention, is, according to Athen. iv. p.
175, a four-stringed instrument, having a sharp, clear tone; cf. Ges.
Thes. p. 935. פּסנמּרין (in Daniel 3:7 written with a טinstead of תand
in Daniel 3:10 and Daniel 3:15 pointed with a Tsere under the )ת is
the Greek ψαλτήριον, of which the Greek ending ιον becomes
abbreviated in the Aram. into ין (cf. Ges. Thes. p. 1116). The word
has no etymology in the Semitic. It was an instrument like a harp,
which according to Augustin (on Psalm 33:2 [Psalm 32:2] and Psalm 43:4
[Psalm 42:4) was distinguished from the cithara in this particular,
that while the strings of the cithara passed over the sounding-board,
those of the psalterium (or organon) were placed under it. Such harps
are found on Egyptian (see Rosellini) and also on Assyrian monuments
(cf. Layard, Ninev. and Bab., Table xiii. 4). סוּמפּניה, in Daniel
3:10 סיפניה, is not derived from ספן, contignare, but is the Aramaic
form of συμφωνία, bag-pipes, which is called in Italy at the present
day sampogna, and derives its Greek name from the accord of two pipes
placed in the bag; cf. Ges.