No. You can't win.
Let's play through the scenario, shall we?
You lynch me today. Now there are 2 cultists alive.
Cryptonic shoots one of us. 1 cultist left. Cryptonic is now out of shots.
Last cultist converts another, 2 cultists alive.
It is now the following day.
You lynch one of us, 1 cultist left.
Night time. Vig's outta bullets, can't kill the cultist.
Last cultist recruits another,
wash rinse repeat until everyone's dead or a cultist.