If the law specifically targeted Catholics then that denomination would not have equal rights to religion. However, I went and looked up the definition just to put the issue to rest:

(18) “Member of the clergy” means any regularly licensed,27 accredited, or ordained minister, priest, rabbi, imam, elder, or28 similarly situated religious or spiritual leader of any church,29 religious denomination, religious body, spiritual community, or sect,30 or person performing official duties that are recognized as the31 duties of a member of the clergy under the discipline, tenets,32 doctrine, or custom of the person’s church, religious denomination,33 religious body, spiritual community, or sect, whether acting in an34 individual capacity or as an employee, agent, or official of any35 public or private organization or institution.

~SB 5375

This is wildly outside of the mainstream interpretation of the first amendment. Whether the law would be upheld by the court is basically a toss up.