So many issues here.
So, you don't have blended families because you cannot stand his child and he wants you to parent her. She has a mother, whether any of you like it or not or whether she is a good person or not. He doesn't get to decide that you "take over" as mom - what in the world does he think that will do to his daughter? He needs to deal with his ex and the issue of parental alienation - and by the way, how do you know what the mom says about you behind your backs? Are you taking the word of a 4 year old? What do you mean about taking a 4 year old to bars? How do you know she does that? And how in the world is it your responsibility to tell another woman what she can do with her daughter when you are not the child's other parent? I'm not defending taking children to bars, mind you - I just want to know why it's your responsibility to get involved in this. What's your legal claim? Why isn't her father getting involved here?
Blended families only work when both people in the marriage are on the same page. You and he are not. Who are you crying/begging/pleading to? The ex? Your man? Why?
Look, she's 4. You prioritize things. You don't take orders from her, but you let her father parent her. You don't worry so much about a 4 year old showing her panties accidentally. That's not first on the list at all.
How were things "before" that you want to "change it back to"? She's acting out for a reason, you say you are "working for months" but you don't say if you are getting professional help in this regard. Based on the way you word your question, there's not much understanding at all about a 4 year old's development or what happens when a family goes through upheaval and a child hears bad stuff about Dad from her mom and picks up on her future stepmother's resentment and her father's total abdication of parenting. If you aren't getting intensive parenting assistance for a child in crisis, get it. And that means your boyfriend takes the lead on it and you absolutely work as a team.
There's nothing in your question that says you are doing this. I'm a stepmother of 2 and I get how incredibly difficult it is, especially when there is resentment and poor communication between father and mother. But what you're doing isn't working - you need expert and ongoing counseling. But it's worth it.
Otherwise this will break up you and your boyfriend - because you have not blended your families at all, you've just put his kid and your kids under one roof.
ETA: Okay, you've put in additional info. Your boyfriend takes screen shots of the child in bars and prints them out. Those go to the lawyer or court. Stuff is stolen at preschool? The school handles that - how is a 4 year old getting to a staff member's purse anyway? Poor supervision. Stuff is stolen from your room? You put your valuables away and you put a lock on the door. Stuff gets taken to Grandma's house? Who is packing her stuff and not going through it? An adult must be supervising a child this age. "Please" and "thank you" - kids don't get what they demand until they ask nicely - learn to ignore a 4 preschooler who makes demands, and have family rules for what everyone has to say/do to get privileges and treats. Same rules for everyone. If she stops getting a payoff for bad behavior, she'll stop the behavior. The rules are the same for your kids.