OP again.. I work part time and DD is only in a morning Montessori program. Our hours coincide beautifully.
If I quit, I would out her in a preschool that has a gradual separation. Meaning, I would stay with her the first day, then the next day I would leave her alone for a half hour, then the next day an hour, pushing it to eventually her being by herself for 3 hours. It will take a long time but I think it will be a more gentle way to get ready for school than what she is experiencing now.
This would take about two weeks, according to to this schedule, right? So, try to get two weeks unpaid leave or have your DH take the time and do it.
This problem your daughter has is a temporary problem, in the big scheme of life and one she will have to get over if she is to go to school. Quitting your job has far larger implications for you and your career. It just doesn't make sense to quit over this.