[mark]

The pressure tank is not needed.

In the diagram I put up, follow the flow. The back pressure across the asov is created when the float valve (connected to output of di) in your brute is closed forcing the diaphram in the asov downward to shutoff flow coming from the third stage. The check valve is required so the asov doesn't reopen due flow in waste line removing pressure. The source of the pressure is from the supply (even with pump off).