You're right that there are differences between library programming and (mere) usage of meeting rooms. But there is nothing against religious programming in libraries and partnering with religious organizations, as long as programming is education rather than proselytizing. Likewise, libraries that display materials from third party groups cannot discriminate against religious groups.
Programming
12. May libraries provide programs with religious themes, or about religious subjects?
Yes, religion is a legitimate focus of programming insofar as it reflects the interests of the library’s community and furthers the library’s mission. The purpose of such an event should be to inform, educate, and entertain rather than to proselytize or promote one set of religious beliefs over other religious beliefs. Libraries should strive to offer programming that reflects the diversity of religious belief or non-belief in their communities and to ensure that there is no perception that the library favors one religious group over another.
13. May libraries collaborate with religious organizations in programs?
Yes. A religious organization should be held to the same standards as any other group that is allowed to participate in a program. As with its own programming, libraries should assure that the programs are informational and do not proselytize, promote, or advance particular religious beliefs or organizations.