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Can you cite a source that Shir Leviim was sung in the Mishkan? (A source not an opinion) As far as I can find, Shir Leviim was first instituted by David HaMelech as the time the Aron Hakodesh was brought to Jerusalem (I Divrei HaYomim 16).
The Shir of the Levi'im the way we know it in Tefilah and the Mishnah at the end of tractate Tamid is from Tehilim and therefore couldn’t have been sung before David Hamelech's time. However it is clear from the Midrash (Bamidbar Rabah Parasha 6, 3; on the Pasuk "Laavod Avodah Beohel Moed") that the Levi'im were singing during the time of the Mishkan before Temple time. (see also ibid 3, 3 ; same you can find in Rashi Chulin 24a DH Yachol)