I'd be tempted to go with "I'm not inviting Molly because she has repeatedly demonstrated that she is unable to behave at my home. " and let the blowout happen. If her friends complained, I'd respond with "If someone else wants to host the party and deal with her behaviour, they can." I don't want to have to deal with trouble makers when I'm hosting a party, and if they are friends with a trouble maker who will flip out if her actions have consequences, that's not my problem.
If you do invite her, though, I'd tell Josh point blank that when Molly starts acting out, he is responsible for tidying her up and taking her away, even if it means he has to miss the party. And *you*, not he, makes the call.
The thing that would really worry me though is if she drives to the party. Because if you don't let her sleep off the alcohol, she sounds like the type who would then say "Fine. If you won't let me sleep over, I'll have to drive home anyways." Then, of course, you have to decide between wrestling the car keys away from her physically, or calling the cops on her, neither of which can be considered low drama.