Can we record our own versions of copyrighted songs to create a compilation for distribution outside of our church, for example as a fundraiser?

To make a recording of copyrighted songs that is sold or made publicly available, you will require the mechanical rights for each song. You may be able to obtain a mechanical license directly from the song owners or via a company called Music Services. You may also need to contact the Performance Rights Organization (ASCAP®, BMI ®or SESAC®) that represents the song(s) you want to use.