Se garder le coup is not an idiom. The meaning can be understood on the basis of the dialogue and what the mayor has in mind.

Se garder means here to keep for later, for one’s personal use.
Ex: je me suis gardé un bouteille de ce champagne pour le Nouvel An.

Coup has indeed many possible meanings, here it should be understood in the vague meaning of an act. A coup can be a clever act in a game (un coup de poker, le coup du chapeau in football, etc.), or a nasty blow to an opponent (un coup bas, Trump a mis un coup à Clinton). A coup is also a shot, with a firearm (fusil à deux coups, tirer un coup, etc), that can also be used metaphorically.
All of these meanings can work here in the context of political rivalries, tricks and low blows.

In any case, the general meaning is clear: the mayor is not willing to use this trick for now, but decides to yet keep it up his sleeve in case they need it later.