This is the first time I've posted on MN, I can't NOT say something to you OP. Long post, sorry.
I know it must be difficult to share your situation and also to hear all of the above. I was in the same situation as you in regards to the relationship, but no children at the time. I cared desperately for the man I had known and been with since we were teenagers. He had underlying issues stemming from emotional abuse/neglect (though he wouldn't call it such) in his childhood. Our relationship was intense and loving except when it wasn't - when he would be angry or 'out of character' when drinking and get into arguments with friends and sometimes low level fights with other people. I was committed to helping him by showing him unconditional love and making him feel secure, eventually making him happier and able to be his better self, that side was amazingly warm and funny and caring. In practice, by extension this also meant ALWAYS thinking about how he may feel about and react to anything and everything happening in his world and trying to influence for the better/mediate to minimise confrontation. He came first for me because that's what love was. It was exhausting at times but I didn't know anything else.
The first time I called him on his behaviour in a non-supportive, mothering type of way he lashed out at me physically. I'm ashamed now to say that I stayed because I was sure it was a one-off, I loved him deeply and was aware that leaving him there and then would have a devastating impact on his self esteem. As it was, after the initial apology he didn't think it was such big deal. I was very lucky to work in the same office space with someone who had for years ran a women's shelter and who patiently got me to see over a series of (what she made seem like light-hearted) talks that the longer I stayed with him and prioritised his emotional development, the lower my self esteem became and that I may not always have the resources to support him never mind myself, unless I 'let him' take charge of resolving some of his issues himself - perhaps without me there to act as buffer he would be more able to admit to/take charge of helping himself with his issues.
Feeling like a terrible person, I left, hoping that we would benefit from time to focus on ourselves and that we'd then be back, stronger than ever. I took care to be as loving as possible even in this. I never stopped loving him in fact but did learn that I shouldn't be with him. A bit of perspective and a painful process of acknowledgement meant we never got back together though we tried again twice. We can all choose how to manage our feelings - I refused to be ok with how he decided to continue to act toward me, issues or not.
The reason I was able/willing to be in this kind of self-destructive relationship in the first place? My own father-daughter relationship. Which sounds eerily like the one you describe between your daughter and her father. My mother had the same reasons you mention for not firmly and definitively stopping him in his tracks - it got worse the older I got - she thought drawing attention to it may make it worse for me, she didn't want me not to have a father, he wasn't physically hurting me. It just taught me how to live with it like she did. Also to be very close to her as though unspoken, we knew we were 'in it together.'
In the last year i had a traumatic birth experience and have been seeing a counsellor. Though unrelated, when the therapist asked me to talk on parenting relationships, she made no bones of the fact that she was quite shocked by some of my recollections of his behaviour and has made me acknowledge that even as a 33 year old, the relationship is still having a devastating impact on my wellbeing. As an adult and now a parent it has also had a negative effect on the way I feel about my beloved mother - I would not allow my child to go through this kind of upbringing and realise that it is a choice she COULD have made - this is heartbreaking for both of us.
Please, please realise that you cannot make him behave differently - as others have said who are suffering from anger issues, your support counts and you don't have to give up on him, I assume you wouldn't want to. However, your daughter is your priority and make no mistake, this behaviour will damage her self-worth and ultimately, relationship with you as well as her father. He needs to be given a wake up call - separation may be it. Hopefully if he does manage to start dealing with his problems he will see that your protection of her in a way as amicable as possible with him, is for the good of all three of you. Good luck, OP, I feel for you.