Acharonim have pointed out that it is difficult to square this with the Yerushalmi quoted by the Chasam Sofer. Why does the Yerushalmi consider the Mishkan to have been a temporary structure regarding Shabbos when building it was a mitzvah? The mitzvah should have defined the structure as a permanent one, thereby prohibiting it to be built on Shabbos.
I would like to suggest another explanation. Rashi (Shemos 39:33) says that no human was able to erect the Mishkan. Since Moshe had not partaken in the building of the Mishkan’s vessels, Hashem had reserved for Moshe the erecting of the Mishkan itself. Moshe said to Hashem that it was impossible for him to accomplish this feat, due to the extreme weight of the beams. Hashem told him to do his share, and He will take care of the rest. Moshe began attempting to erect the Mishkan, but it erected itself.
It is difficult to suggest that an individual has desecrated Shabbos by merely attempting to perform an action that Hashem miraculously completes. Since Rashi tells us that Moshe was unable to perform this action and that Hashem instructed him to only make the attempt, we cannot consider that this was problematic on Shabbos.