[LAU] Video micing question

Sorry this is a bit OT, I don't know whom else to ask. I belong to a martial arts community which produces videos of guest teachers when we host a seminar. We used to use a lavalier mic on the teacher's chest. For reasons I don't understand, that practice was stopped, and the mic was moved to the high ceiling rafter, where it picks up the whole room. The result is that some of the most revered figures in our art have had their words lost in the background noise. If I can't convince those in charge to use the lavalier mic as it's designed to be used, would a camera-mounted shotgun mic do better? Any suggestions as to model? Any other suggestions for micing a moving subject 30 feet or so from a wall-mounted camera? I will try to get more particulars about the setup. I have noticed 2 XLR connectors on the camera which are not used. The mic and the camera are fed separately into the computer, I don't know what happens from there. Probably uses Final Cut Pro. Thanks, Paul