It depends on whether Alice is a "minister."
This case plays on the tension between employment-discrimination laws that prohibit discrimination on the basis of religion, and the First Amendment, “which protects a religious group's right to shape its own faith and mission through its appointments.” Hosanna-Tabor Evangelical Lutheran Church & Sch. v. Equal Emp't Opportunity Comm'n, 565 U.S. 171, 188 (2012).
We resolve the question by asking whether Alice is a "minister," i.e., whether the religion has entrusted her “with the responsibility of educating and forming students in the faith” Our Lady of Guadalupe School v. Morrissey-Berru, 140 S. Ct. 2049, 2069 (2020). If so, she falls within the First Amendment's "ministerial exception" to employment-discrimination laws.
But the precise definition of that term remains rather squishy, so we don't really have a test that can give us a reliable answer. In Hosanna-Tabor, for instance, a teacher at a religious school brought an ADA claim after the school fired her after she missed a semester on disability leave. The court rejected her claims based on the religious character of her work, and the Supreme Court affirmed, concluding that she was a minister based on the job title she received (Minister of Religion, Commissioned), her own use of the minister title, the religious character of many of her job responsibilities, her religious training, and other factors.
Although she was a minister who was replaced by another minister, the Court suggested that the outcome might be the same if she were not a minister. Noting that the school in that case – just like in your hypothetical – hired lay teachers only when ministers were not available, it rejected the argument that an employee falls outside the ministerial exception simply because there are non-ministers with the same job functions:
Though relevant, it cannot be dispositive that others not formally recognized as ministers by the church perform the same functions—particularly when, as here, they did so only because commissioned ministers were unavailable.” Hosanna-Tabor at 193.
And a few years later, the Court indicated that perhaps none of those factors from Hosanna-Tabor are actually required. Instead, it directed courts to ask whether the school “entrusts a teacher with the responsibility of educating and forming students in the faith.” Our Lady of Guadalupe School at 2069.
So in your hypothetical, the courts would probably look to see whether BYU treats its professorships as ministerial positions. Although the answer could vary from one religious school to the next, it seems quite likely that the courts would treat BYU professors as ministers, given their Personnel Conduct Policy:
It is a condition of employment that all personnel (including student employees) act in accordance with university policies and the Church Educational System Honor Code, including the Dress and Grooming Standards, and refrain from behavior or expression that seriously and adversely affects the university mission or The Church of Jesus Christ of Latter-day Saints. Examples include behavior or expression that
- Contradicts or opposes, rather than analyzes or discusses, fundamental Church doctrine or policy;
- Deliberately attacks or derides the Church or its general leaders; or
- Violates the Church Educational System Honor Code.
All personnel are expected to be role models of a life that combines the quest for intellectual rigor with spiritual values and personal integrity, and to conduct their work in a professional manner consistent with the values espoused by the university and the Church. Members of the Church in nonstudent positions also accept as a condition of employment the standards of conduct consistent with qualifying for temple privileges. The university regularly contacts ecclesiastical leaders concerning the temple eligibility of all nonstudent personnel who are members of the Church.
I couldn't find any cases applying the ministerial exception to Brigham Young, but given its expectation that employees act as "role models" for the church's "spiritual values" and conduct their work in a manner consistent with "values espoused by ... the church," I would be surprised to see a court get in the middle of the school's decisions to fire Alice and replace her with a member of the church, even if she had not run afoul of those policies herself.
The interest of society in the enforcement of employment discrimination statutes is undoubtedly important. But so too is the interest of religious groups in choosing who will preach their beliefs, teach their faith, and carry out their mission. When a minister who has been fired sues her church alleging that her termination was discriminatory, the First Amendment has struck the balance for us. The church must be free to choose those who will guide it on its way.
Hosanna-Tabor at 196.
Still, Alice could have other bases for suing the school. Based on your tags, I'm assuming you're most interested in employment-discrimination claims. While those might be barred by the First Amendment, Hosanna-Tabor explicitly held open the possibility that a contract claim might still be viable. Hosanna-Tabor at 196 (“We express no view on whether the exception bars other types of suits, including actions by employees alleging breach of contract or tortious conduct by their religious employers.”).
So if the school signed a contract agreeing to employ Alice for three years, she would have a strong argument that it had waived its right to discriminate on the basis of religion for that period.