Could modern unmodified Airbus aircraft be used for zero-G flight?
According to this source and this source, the relevant A320 control laws (CLAWS) limitations are:


Normal load factor limitation in clean configuration down to -1g
Nose down pitch attitude down to -15deg
High Speed Protection (HSP) to prevent exceeding VD/MD


The problematic one would be the pitch attitude limiting. Since the pitch attitude will keep decreasing at zero G, the limiting function will pull the aircraft up and increase the load factor after some point. This could limit the time in zero G below the requirement. 

Given the constraints and objective, more analysis would be needed to ascertain whether aircraft in Normal Mode is acceptable.