I don't feel that guilty about the LIS lynch, because if he's really wanting the villagers to win, he knows we have to lynch him.
My only concern is that lynching LIS gives the convert a free night. If the convert gets another conversion this game, a free night likely puts them closer to their conversion chance. Wouldn't it be a better idea to leave the LIS lynch until we think we've got the convert? Or does that risk letting LIS win the game while we're still hunting the convert? I'm not too familiar with neutrals, so I don't know if a neutral win occurs at the end of the game, or if it can end the game early.