S.
I understand your situation. I have been taking my son to supervised visitation with his father for 18 months now. I know the draw to let your child know her father. At court, the thing that most impacted me was one of the judge's statements. "A child is best with two healthy parents." When one parent isn't healthy, the child is best with the healthy one. Your ex-husband doesn't sound healthy.
You'll weaken your court orders if you let them be alone--even if for only 2 hours at the mall. Even when it's on paper, if you haven't been following it; it can be challenged and more successfully if you allow variations. An objective third party agrees that your daughter should not be alone with your ex. It's not just you that thinks so, even though he likely claims it is.
One of the stipulations a local psychologist who does custody evaluations wanted in any parenting agreement in our case was: "father should perform all care taking during visits". If your ex works 12 hour shifts, then sleeps even say 6 hours rather than 8--your daughter only has 6 hours with him. That's if she isn't sleeping.
You've been remarkably accommodating to him, BTW.