Of central importance to this post was the idea that Rashi held that Moshe's service as kohen gadol during the seven inaugural days were considered as service in the mishkan itself. However, Rashi at the very end of Parshas Pikudei (40:29, see also Rashi to Pasuk 31) says something very strange. The pasuk says that Moshe brough the olah and mincha on the mizbeiach. Rashi says "Even on the eight day, the day of erecting the Mishkan, Moshe served [as Kohen Gadol] and brought the communal offerings except those [special korbanos] commanded only for that day...." So he actually served as Kohen Gadol on the eighth day, when it certainly had the status of Mishkan.
This contradicts what Rashi says in Zevachim 19b s.v. Moshe, that Moshe and Aharon did not serve as Kohen Gadol at the same time. There are other Rishonim who say that Moshe served as KG on the eighth day, see Shita Mekubetzes, Zevachim 19b 3 in the name of the Ra"m.
The Netziv in Haamek Davar, 40:23, says that there were two versions of Rashi and he changed his mind; I guess at first he felt Moshe served on the eighth day, then he changed his mind that Moshe did not serve on the eighth day. (I was not sure what the Ramban was saying in Rashi.)
See also Torah Shleima 47 and 55* at the end of Pikudei about Rashi.
There is a third opinion as to the duration of Moshe's service as Kohen Gadol. The makshe in Tosfos, Avodah Zarah 34a s.v. Bameh holds that for all 40 years (until his death) Moshe was considered Kohen Gadol. He bases it on Zevachim 101b: Rav said, Moshe was kohen gadol and took a portion form kodshai shamayim. The Gemara does not say explicitly that he was kohen gadol until his death, but that is how the maksheh in Tosfos A"Z learns.
Maybe Moshe only served as Kohen Gadol for the seven days, or even the eighth, and he never served again. But once he served, he retained some special status. (Whether people in Galus [for killing accidentally] needed to wait for Moshe's death to be free, I do not know.) Maybe the makshe in Tosfos in A"Z doesn't mean Moshe was actually the Kohen Gadol the rest of his life, but he had some special status that must have manifested himself in certain ways, like in his dress. See Shiurei Rabeinu Meshulam David Halevi Zevachim Dapim 2-25 (Jerusalem 5767, edited by A.C. Shpiegel) p.387 who brings Rav Chaim as mentioned by the Brisker Rav something along these lines: Moshe served for the seven inaugural days but that doesn't mean he lost his status after that; he was not a zar (non-Kohen) and needed to wash from the kiyor when he entered the Mishkan.