Gosh, you sure get up early! All I can say is this is my 1 time in thousand to disagree with you and be right. I have a funny feeling I better appreciate this one because it will be a lllooonnnnggggg time before it happens again.
What do you think about using a PRV on the church? Other than putting pressure tanks behind your checkvalves, which the other guy said would create a new set of problems, I can think of no other answer.
Of course, I still don't understand the "ping-pong" thing. I would think the large tank would basically act like a pump to supply water to the two smaller tanks and the two checkvalves would prevent oscillations. But, maybe not. The other guy seemed pretty certain.