Rashi writes (VaYikra 9:23) that after the miluim, after all the korbanos, the Shechina still did not come down to the Mishkan. Aharon blamed himself, thinking that it was his role in making the eigel that prevented the Shechina from descending. He came to Moshe and complained and asked Moshe why he ever put him up for the job of kohen gadol only to face public embarassment and humiliation. Moshe responded by immediately entering the Mishkan with Aharon and davening with him until the Shechina descended.
Rashi further writes that the people saw how for seven days Moshe put up the Mishkan, performed the daily avodah, took it down. When the Shechina still did not enter the Mishkan, the people came to Moshe humiliated, wondering if their efforts in making a Mishkan were in vain. Moshe responded by telling them that once Aharon does the avodah the Shechina will descend, demonstrating that Hashem accepted not only the Mishkan, but accepted Aharon as kohen as well.
R' Baruch Mordechai Ezrachi in his sefer Birchas Mordechai says a beautiful vort here. The fact that the Shechina did not enter the Mishkan for seven days did not bother Moshe. The fact that the Shechina did not come down immediately after Aharon did his avodah did not bother Moshe. The Shechina itself seemingly was not moved by the construction of the Mishkan, the offering of korbanos, the service of Aharon. What changed things? What caused the Shechina to enter the Mishkan? It was Aharon's feeling of embarrassment that changed things. It was Klal Yisrael's feeling of humiliation changed things.
Had the issue been purely a bein adam laMakom, there might have been very good reasons for the Shechina not to come down to the Mishkan. Who knows what was needed for kapparah? However, when the Shechina felt the pain of Klal Yisrael, the suffering of Aharon, it relented and was drawn to the Mishkan to comfort and be close to Klal Yisrael. When Moshe heard that there is a bein adam l'chaveiro issue
here, that his brother was suffering embarrassment, that Klal Yisrael
felt humiliated, he stuck his neck out to beg for rachamei shamayim and Hashem responded.