In his commentary on on II Kings 12, which deals with King Joash's temple repairs, T.R. Hobbs argues that the chapter could not have come from a priestly author:
The record of the events undoubtedly has a Judean origin, but it is highly unlikely that it can be attributed to priestly circles. The Kings account of the temple repairs, compared to that found in 2 Chr 24, is decidedly critical of the priesthood. The blame for the lack of repair to the temple after Joash's original order is laid upon their shoulders ... and, as a result of that failure, new regulations are introduced.... In the procedure outlines by those new regulations the priests play a minor role. In 2 Chr 24:4-7 it is the failure of the Levites, not the complete priesthood, which is noted.
The reform also takes place at the initiative of the king, not the priesthood, nor one of its members such as Jehoiada. The king's accountant actively participates in the reform ..., and the priests themselves are eventually refused access to a source of income they had hitherto enjoyed ... because of their failure to obey the king.
The tone of the chapter then is far from "priestly," and is in fact highly critical of the priests.
The background to this is Wellhausen's hypothesis that the biblical narratives came from a number of hands across several centuries, with one source specificially coming from a priestly hand in order to promote priestly agendas in opposition to the other agendas of biblical authors that conflicted with priestly concerns. Wellhausen's specific proposals have largely been rejected, even if some of his structure has been retained by many critical scholars. None of them can agree on any of the details, which is some reason for wondering if the whole thing should be rejected as thoroughly unfounded, but the fact remains that many scholars accept something of Wellhausen's source proposal. As for this chapter, I wonder if we can really say much about the authorship. Hobbs overstates his case in several places.
For one thing, the priests are not a larger group than the Levites, as if pointing to them means a smaller subsection of the priests. The reverse is true. Sometimes the Levites are treated as a contrast with the priests, and so the context makes it clear in such cases that the non-priestly Levites are in mind. There is such a reference to "the priests and the Levites" in the Chronicles parallel that Hobbs refers to. But the Levites are the broader category, and failing such a context we might just as easily take a reference to the Levites to include the priests, who were indeed from the tribe of Levites. There is some reason to take the Levites in Chronicles as the non-priestly Levites, given that both are mentioned earlier, but it could just as easily be a shorthand to reference both groups mentioned earlier, since the priests are Levites. If so, then the Chronicles mention of Levites rather than priests might simply indicate that all those who served in the temple were responsible, and the Kings reference to priests might reflect the fact that the priests were ultimately responsible for whatever happened in the temple, even if non-priestly Levites were partly at fault.
But that isn't something I'd rest a lot on. Even so, I'm not following the reasoning here. Is the argument supposed to be this?
1. This passage is critical of the priests at this time.
2. No priestly writer would ever be critical of priests at any time.
3. Therefore, this passage couldn't have been written by priests.
The second premise is undoubtedly false. Why couldn't a priestly writer be critical of other priests, even an entire generation of them? Perhaps this is what is meant:
1. This passage is critical of the priests at this time.
2. Wellhausen's supposed priestly source P is never critical of priests at any time.
3. Therefore, this passage couldn't have been written by the supposed priestly source P that Wellhausen concocts out of thin air.
That argument I can agree with. Wellhausen proposed a certain agenda for his priestly writer(s). But Hobbs says that it's "highly unlikely that it be attributed to priestly circles". I just don't see how that follows from the conclusion of this argument. It's not likely to be from a circle that uncritically seeks to justify or excuse everything the priests might ever have done, but I'm not sure any of the biblical texts come from anyone like that. I certainly don't think Wellhausen's supposed P author, if that's an essential characteristic of that source, could have produced the biblical material we actually have, unless a later editor thoroughly reworked things to remove such assumptions.
I'm open to source theories of composition, especially in Samuel and Kings, where the text tells us that sources were used. But too often we get this sort of argumentation passing for scholarship with rhetoric about scientifically determining what those sources must have been and what the agendas of the various source traditions were. Sometimes you even get some kind of nonsense about the final editor stupidly placing contradictory materials side-by-side as if the final editor had no clue or no care about the supposed theological contradictions going on. Or you might instead get some proposal of an agenda being portrayed by the narrator that conflicts with the agenda in one of the sources, and the editor comes out as someone who must have been pretty poor at masking alternative agendas in the source materials. I have a hard time calling this stuff serious scholarship, since there have long been ways to deal with supposed contradictions that can just as easily deal with supposed conflicting agendas. Literary scholarship in recent decades has shown how anachronistic, Western-centric, and uncharitable such readings are.