I don't know where you got the idea that the Catholic church has taken no position on Santa Muerte. The current official position of the Catholic Church is that honoring Saint Death is heresy.ref Mexico's Catholic Bishop Conference has accused Santa Muerte devotees of mixing Christianity with devil-worship.ref According to a statement by Roman Catholic Archdiocese of Mexico City on November 3, 2008, popular devotion to the so-called "Saint Death" is not compatible with the Catholic faith.ref The U.S. Conference of Catholic Bishops has not issued an official position on this because this is a relatively new phenomenon in the United States.
What would it take for them to do so? If a priest were to acknowledge her, would that be sufficient?
A single priest's action would not force the universal Catholic church to take a stand. If a priest, who holds a position which is responsible for souls, acknowledges her as a saint, then the local bishop will have the duty to correct him. If the local bishop fails to do so, then the pope with the Roman Curia will interfere and instruct the bishop.
How would the church change as a result?
With regard to dogma, faith, and related matters, just as for the past 2000 years, the Catholic Church is not going to change. Not for Santa Muerte, or anything else for that matter. But I have no idea about the cultural consequences. The Church might be persecuted, but it is not going to change its teaching: honoring Saint Death is heresy.
Note: Traditions like Our Lady of Lourdes and The Lady of Guadalupe are about a real person, Blessed Virgin Mary, the mother of God. But Santa Muerte is not a real person, nor do her followers claim as such. So your comparison is not correct.