reversing DC polarity after each use
What can I put in the oval in the diagram to make the polarity switch
on the battery?

There is nothing effective that can be put in the oval area that will work. IRFZ44N MOSFETs (if that is what you mean) are not bi-directional. They will heavily conduct in the reverse direction due to their bulk diode.
Neither is it sensible to connect the faucet sensor directly to the gate of a MOSFET. These are electrostatic sensitive devices and can be easily destroyed without protection circuits.