As anyone familiar with this story also probably knows---and by anyone, I mean you who are reading this post right now---BYU took the position that its police department is a private agency of the university (which means ultimately of the LDS church), and therefore is not subject to public scrutiny of its investigative records.
Funny thing about that: it seems that when laws applicable to governmental agencies are unfavorable to BYU, its police department is claimed not to be a government agency. But when BYU stands to benefit from laws applicable to governmental agencies, suddenly its police officers are government employees. In 2014, BYU took the latter position in front of the Utah Supreme Court. People who hold to a concept of consistent reality might notice that this is the exact opposite of what BYU claimed in the GRAMA case.
The 2014 case was called Mallory v. Brigham Young University, 2014 UT 27, 332 P.3d 922. A PDF of the Utah Supreme Court's ruling is here. If you would rather see it as HTML instead, it's here.
In Mallory, the eponymous plaintiff went to a BYU football scrimmage and left on his motorcycle after the game. A BYU traffic cadet working for BYU's police department was directing traffic under the supervision of an officer from BYU's police department.

this guy Mallory got into an accident with a car and said it was the BYU traffic cadet's fault because she screwed up directing traffic. So he sued BYU. But BYU argued that it was protected by the Utah Governmental Immunity Act.
The Utah Governmental Immunity Act is a statutory scheme (in this context that means "framework," not "trickery"---although the latter is debatable) that sets limits on when and how people can sue state and local governments and governmental agencies in Utah. You may have inferred that from the name. One of the provisions in the Act, which is especially pertinent here, is that people seeking to sue the government or a governmental agency have to send a notice of claim to the agency or its designated agent prior to filing suit. The agency then has a period of time to pretend to evaluate the claim and either settle or deny the claim. If a plaintiff does not follow this procedure before filing a complaint in court, under the terms of the Act their lawsuit will be dismissed.
So BYU accordingly said in Mallory that its police officers, including this traffic cadet, were operating under the authority of and supervision of the Provo police department. Therefore, according to this argument, the BYU police department is composed of government employees who are shielded by the Utah Governmental Immunity Act, and since Mallory didn't send in a notice of claim before filing suit, the courts had no jurisdiction to consider his case and had to dismiss it. 'Cause he didn't follow the rules. And the majority of the Utah Supreme Court agreed with this argument and ruled that BYU's police officers were government employees. So Mallory's case against BYU was dismissed.
Note: it's not that BYU's police officers were independent contractors performing a governmental function (directing traffic). It's that its police officers were government employees, full stop. From Chief Justice Christine Durham, ¶ 21:
In light of this well established standard for determining the existence of a master-servant relationship, we hold that the BYU Defendants were servants—and therefore Employees—of Provo City because the city retained the right to control the manner in which the BYU Defendants directed traffic. Support for this determination comes from the Provo City Code, which strictly regulates the circumstances under which the BYU Defendants may perform the city's nondelegable duty of traffic control, and contains several additional controls over the direction of traffic by nonpeace officers, including the city's ability to terminate the BYU Defendants' service at any time.
TL;DR: That which is a fact under one circumstance, may be, and often is, not a fact under another.
Now, there is a way you can look at this in the best light possible for BYU and say that the issue isn't whether its police department is a governmental agency in general, but rather whether it is acting as a governmental agency in a particular circumstance. Unfortunately, even though one might make a straight-faced legal argument to that effect, in terms of the totality of the real world, that's the worst light possible for BYU. I'll devil's advocate this below, because "devil's advocate" is a verb.