I have just been listening to the Oct 14th radio show where y'all discussed this letter. I think you are correct in your approach; you are not for profit, and they MUST tell you what specific complaint they have or you cannot respond. It is very likely that none of the publishers in their organization have any lyrics listed on your site; make them prove you have erred in this way! You probably DO NOT NEED TO COMPLY WITH THEIR DEMAND, even though they claim you do, because you are nonprofit. Because there can be no claim that any profit is being gleanned from these posts.
I suspect that this organization, like the others, is just trying to scare organizations like yours into buying yearly renewed memberships ... they are playing a numbers game. They have no real power, and may serve no real purpose!