Designing a linear analog filter is not the best answer to this question(!)
It's very surprising if the current taken by the piezo sounder is causing the problem because the current taken by a piezo element is typically less than 4 mA.
It's not the amount of current it takes, it's the digital switching that causes noise that can be conducted to the rest of the components.
but there must also be a resistive component to its impedance.
The capacitance of a piezo element is typically about 15nF but there must also be a resistive component to its impedance. Even if a piezo element is driven with a square wave, it shouldn't be difficult to provide adequate decoupling.
The problem is the mechanical bounce when the power is removed, you can get nearly 100V so I would try a reverse biased diode like you do for motors or inductive loads.
Flyback diodes work because the inductance causes the voltage across the load to reverse when the current supply is opened, which forward biases the diode and provides a path to dissipate the energy.
Have you ever seen the output of a piezo when you give it a wack? It is both positive and negative, it is based on the crystal deformation. If you have a DC voltage the crystal deforms, then when it is removed the force is removed and the crystal is restored, thus generating the reverse voltage. Exactly like a coil.
You want something like this: