BuddyPress.org

== Clearly the problem is with the themplate. ==

You could be right. That theme has had different incarnations though. So are you using the stand-alone Church theme and installed the BP Template Pack to make the Church theme compatible with BP? Or, are you using the Church child theme of Genesis and installed Genesis Connect for BP Compatibility?

To isolate the problem furthere, did you deactivate all plugins except BuddyPress (and BP template pack or Genesis Connect) and see if you can log in?