Yes, a local Catholic church can have an LPFM (Low-Power FM) station even if the Archdiocese has full-power broadcast holdings. LPFM stations are typically considered separate entities from full-power stations and are subject to their own set of regulations and licensing requirements.
The fact that the Archdiocese, a higher-level organization, holds full-power broadcast licenses does not necessarily preclude a local Catholic church from obtaining an LPFM license. LPFM stations are often intended for hyper-local community broadcasting and may have different ownership rules compared to full-power stations.
It can be considered a separate "chapter" of the larger organization as long as the church is separately incorporated or otherwise recognized by the state and has a distinct educational mission. (see 73.858 of the rules)
However, it's essential to ensure that the local Catholic church meets all the eligibility criteria and licensing requirements set by the Federal Communications Commission (FCC) for LPFM stations. This includes compliance with non-commercial and educational broadcasting rules, among others. Consulting with FCC specialists or legal experts can help ensure that the LPFM station is set up in accordance with all applicable regulations.