We can save the bitten player from death by taking advantage of the fact that the Mafia controls their vote. We execute the Non-Neutral Villain (NNV), but don't add one vote. The Mafia won't be able to hold back and, in order to hang the NNV, they will add a vote from the bitten player. We will find out who the bitten player is and heal them during the night! Profit!
Besides, we have double votes.