OP, I think you have had a very hard time on this thread.
You say in your OP that you know that in the occasion you were violent you were wrong, you have never forgiven yourself and you later told us you were then diagnosed as bipolar.
Does your Dd know that you yourself have MH issues?
The thing is that you are her Mum. Good, 'good enough', 'trying hard' you are the only Mum she has and you are clearly trying to help your dd.
I presume you have told her your treatment on that day was wrong, and you are unreserved sorry?
But the thing is, you can't be strong for her if you are constantly crumbling through self doubt and guilt. Are you getting any help? Talk to your GP.
Any number of things may be going on. Have you talked with your dd about how you felt, growing up with in-dx'd bipolar?
Who knows , she may have inherited MH tendencies, she may be testing you: making accusations to see if she can push you away, the insecurity of the self fulfilling prophecy. 'I'm not lovable and I can prove it because if I do or say this my Mum will desert me'.
I have seen so many people with self harming traits do that.
Your job, of course, is to hold fast, and let her know that nothing she can say or do will shake your unconditional love. She may play out behaviour you don't like but you still love her and will not reject her.
You need to be firm and strong to do this. Be clear about your feelings about the past and then be strong and supportive for the present.
But I do think help for you might help you be strong for her.
Good luck.