AITAH for not approving of my daughter's boyfriend and not letting come over to the house.

I (F44) have my daughter (25) still living with me. She doesn't work. She has two young boys (7 & 4) that I have been helping her raise and supporting. She has BPD which makes her prone to lashing out and can make life difficult. But we manage. She started dating this guy 5 months ago (M34). He was not to be around the boys for 6 months. He understood this. However, he came over to the house twice when the boys were up. He walked right into the house and went up to her infront of them. She got upset and they argued (away from the boys). And again he 'understood'. He then tried to show up at the boy's sports practice. Again they talked and he finally got it. They didn't spend a day away from each other. Every night they spent together and she would sneak outside to see him during the day. The eldest boy figured out something was going on. So we ended up telling him and they were introduced to BF at around 8 weeks in. And then he was at the house even more. Sometimes 3 times a day and every night. I find the PDA too much, he has to be touching her even in front of me. He asked to sleepover around 3 months of them together. I said no. He wasn't pleased with that answer and asked a month later, I still said no. Month 4 was difficult, they started arguing and she was crying almost nightly for a week straight. He had her get Life360 so he could track her and he was watching it ALL the time. He would call and accuse her of being out when her and I were watching a movie or not at the school picking up the kids. When we traveled for the boys to see their father, he accused her of cheating on him with their father. He demanded she walk to his place that night when she was in pain and could barely walk, she asked him to come over and he said no because he didn't want to talk with me there. I do not intrude on their time. And our house in large enough that they could have a private conversation without me interrupting. Two days later he told her she was being shady because she told him that she was taking the boys to see a movie with a friend and her kids. This was my last straw with him. He came by the house a week or so later to tell me he was going to stop doing these things and 'be better' though he could not tell what it was he was changing or how he was going to be better. Spoiler, he did not. Her half-sister came to spend the week and every single night she sees the BF. Her sister does not always join them and I entertain her or she sits in her room on the phone or watching TV. He continues to track her movements and question her whereabouts. He requests her to sleepover. But she doesn't bring it up with me until 1 boy is in bed and the other is going to bed. Her not being home when they wake up would upset them, she knows this. And they wake up at 5am. I said no. She cried and argued with me for 2hrs. Saying he'll break up with her if she doesn't go. I finally relented. She left for his place around 11pm. 2:51am I wake up to her screaming to let her in (I put the chain on the door in case the kids tried to go outside looking for their mother when they woke up). She is hysterical, crying and shaking saying he broke up with her because he's tired of 'fighting for her attention.' She pleads with me to make it better. We sit at the kitchen table and talk. I get more details, she was worried about the boys (one had not been feeling weĺl) and she was feeling guilty for not being home. He was frustrated and upset. And it ended with her coming home and him saying that. I explained any man that demands you put him above your kids is not worth your time. She's a single mom and the only consistent parent. I told her that she deserves better. Fast forward 24hrs and they are back together and I'm completely stunned that she even forgave him. Though she says she hasn't but they are working on it. It's been over a week and she's begged me to give him yet another chance. But I told her no. AITA for not giving him another chance? I think that if this much has happened in 5 months that it's not going to get any better. Any advice would be much appreciated.