They are lynching her because she is a dark elf. That is how drow were treated in FR. In bg1 you have to fight a Flaming Fist as well to recruit her. It makes perfect sense in the setting.
Neither situation makes you lose reputation though. You get a -2 rep penalty when she joins the party (it goes away when she leaves too) which is again representing the setting with drow feared and treated with suspicion, disdain and outright hate.