Mushite priests vs Aaronid priests in the Bible.

In Vayikra, the point is often made that the only valid priests are the descendants of Aaron. This is made clear with the formula בְּנֵי אַהֲרֹן הַכֹּהֲנִים or הַכֹּהֲנִים בְּנֵי אַהֲרֹן.

The theory of the critics is that Leviticus is the work of Aaronid priests who were protecting their prerogatives. They wanted it made perfectly clear that they, and no one else, were the true priests.

As noted by the critics, outside of the material attributed to the Aaronid priests (specifically Deuteronomy) the priests are often called הַכֹּהֲנִים הַלְוִיִּם, which say the critics, suggests some sort of rivalry between the Levites, or a family of Levites, who thought they were entitled to serve in the Temple and the Aronids who wanted it made clear that right belonged to them alone.

But is there any other evidence of such a rivalry?

Yes, some.

When David first establishes himself in Jerusalem, he has no Temple, but he does have two high priests. This is stated explicitly in 1 Chronicle 15:11, and 1 Kings 4:4 where Abiathar and Zadok are named as co-high priests.

Who were these men?

About Zadok, very little is known. He appears out of nowhere in Samuel, but Chronicles identifies him as a descendant of Eleazer. We know far more about Abiathar.

Unlike Zadok, who is given no clear background story, Abiathar, a survivor of the massacre at Nob, is depicted as David's loyal friend during the dispute with Saul. Abiathar possesses the Ephod and uses it to answer David's questions.

Abiathar is also linked to Eli explicitly in Kings 2:27, and the connection is hinted at in other places.

Why did David need two High Priests?

Several answers are offered, but we will confine ourselves to the one that relates to this discussion.

Frank Moore Cross suggests that there were two religious cults at the time, one controlled by Aaronid priests who traced themselves to Aaron, and one controlled by Mushite priests who traced themselves to Moses.

About Zadok nothing is known, but Abiathar is a descendant of Eli.

In 1 Samuel 2:27, Eli receives a very strange prophecy.

"Now a man of God came to Eli and said to him, “This is what the Lord says: ‘Did I not clearly reveal myself to בֵּית אָבִיך when they were in Egypt under Pharaoh?"

As Chazal knew, God never reveals himself to Aaron in Egypt: the only revelation is to Moses. Moore takes this as evidence that Eli is a descendant of Moses, and a Mushite priest, while Chazal say this verse is evidence that Aaron did, indeed, prophesize in Egypt, though no such prophecy is recorded.

If we accept the suggestion that Eli, and his descendant Abiathar, were Mushite priests the answer to the question snaps into place. David, the shrewd politician, wanted to be seen as legitimate by people who worshipped with the Mushites and by people who worshipped with the Aaronids. Choosing one denomination over the other might be asking for trouble, so took a high priest from each group, and this arrangement continued until Abiathar backed the wrong heir, and was fired by Solomon

Is there any other concrete evidence of a Mushite priesthood?

Yes...

• Judges 18:30 describes a priesthood in Dan, established by Mushites. The verse reads "Jonathan son of Gershom, son of Moses, and his sons were priests to the tribe of the Danites until the time the land went into captivity." which may suggest that these Mushite priests of Dan continued to serve at Jeroboam's temple in Dan.

Is there any other less concrete evidence of a Mushite priesthood?

Yes...

• In Exodus, Moshe and Joshua act as priests.
• The old poem attached to the end of Deuteronomy seems to identify Moshe as the founder of a priesthood:
  
  "About Levi he said: "Your Thummim and Urim belong to your faithful servant. You tested him at Massah; you contended with him at the waters of Meribah."
  
  Moshe is the faithful man, of Levi, who was tried at Massah and Meribah. The verse says the icons of priesthood belong to him as well.
• There are several stories of conflict in the Torah, that may be interpreted as polemics against one of the two priesthoods, for example:
  
  a) In the Golden Calf story Aaron is the one who makes the young bull, and a bull is what was used in place of cherubim in the Bethel Temple. This may be coded criticism of the Bethel rites, which would have been administered by Aaronids. At the end of the story, Moses and the Levites go to battle against everyone corrupted by Aaron's act and as a reward, they are "consecrated to the Lord" and given a "blessing".
  
  This story also tells us the Levites were congratulated for "slaying their own brothers, companions, and relatives" and, in Deuteronomy, the old poem mentioned above suggests the Levites were rewarded with the priesthood because they "didn't recognize their brothers or children"
  
  b) In Numbers, the Bal Peor Heresy occurs "before the eyes of Moses" who seems powerless to address it. He fails to act. Instead, Pinchas, the Aaronid, saves the day, and for his actions, he is rewarded with "eternal priesthood" The text can be read as suggesting the priesthood passed to the Aaronites because they cleansed Israel from a Midianite taint, and Moses, of course, is married to a Midianite, and the father of half-Midianites. Eli, it must be remembered, also loses the priesthood because his two sons, like those who were guilty of the Baal Peor heresy,  "lay with women at the door of the Tent of Meeting"
  
  c) In Numbers 12 Moses is attacked by Miriam and Aaron for marrying the wrong woman (she is identified as a Cushite in this story, but Moses married a Midianite) but the issue becomes "Has God ONLY spoken to Moshe? Hasn't he spoken to us, too?" God answers by telling Aaron and Miriam that Moses outranks them both. The theme seems to be that Moses is superior to Aaron, despite the "mixed" ancestry.
  

Does this "prove' anything?

Of course, not. There are no proofs in interpretation.