Christian Copyright Licensing International (CCLI) has recently partnered with Motion Picture Licensing Corporation to offer a service especially for churches. Church Video Licensing (CVLI) offers an "umbrella license" for an annual fee, that works a lot like your CCLI license to print or project song lyrics.
You can show authorized titles from MPLC's long list of big name studios (including Sony, Warner Bros., and Disney) and from a growing list of Christian movie producers. This license covers clips in sermons, plus videos shown in classes, youth groups, and events such as family film nights, so long as no admission is charged and the title is not advertised to the general public. Fees are based on church size. Licenses can be granted over the phone. For more information, contact CVLI at 888-771-2854 (CVLI) or visit their website.