Overkill or wrong tool idiom
Googling "using a * to kill flies" (with a wildcard) yields, in the first two pages of results,
hammer,
bazooka,
shotgun,
sledgehammer,
flamethrower,
bullet,
cannon,
battleship,
machine gun,
howitzer,
tank,
as well as flyswatter. So it's clear that using X to kill flies is an established idiom, but there's no single word that is predominantly used for X. Pick your favorite weapon or tool and insert it.