I second what most have said. He should NOT be in the picture. However, I may have some unique perspective for why I say that, due to my various forms of volunteering.
Without going into all of the ways I volunteer, as some may distract from the answer, they key point is that I volunteer quite a bit with children, and in particular I often volunteer with single mothers by choice (people who choose to have a child while single).
The first thing I should say is that single mothers can be great parents! I know you hear all these statistics about how kids raised in single parent homes tend to be less adjusted and stuggle more and these statistics are technically accurate, but their also very misleading. The reason that kids struggle isn't so much that the mother's are single, but how they became single. Most kids in single parent homes have young (teen) moms or parents who didn't want a child at all, basically the parent in question was not ready/willing to be a good mother. When you look at single mothers by choice your see that they can do just as good a job raising a child as a couple can, I've known many amazing girls of single mothers, I just spent last night babysitting one and she is not just happy and well adjusted, she is smart, active, very self assured, and her biggest problem if anything is being too friendly (she walks up to strangers and acts like their her best friend lol).
My point being that a father figure is not needed in a child's life for them to do well. Sure if they are a good father figure that is great, but if they are not a good role model they will do more harm then good. I've see this many times as well working with so many children, kids who are doing well alway who seem to 'regress' into bad behaviors after a visit with the other parent because of the poor behaviors modeled by the parent. If a parent is not going to be a generally constructive and positive role model for a child the child will be better off without them in general!
Now lets look at this father, he is not just a bad role model, he is a horrible one!!! For starters you do not deserve what he is doing to you, and should not have to tolerate it for the sake of your child. However, it's bad for your child as well for him to be present. Best case he would be teacher her bad habits and not providing anything constructive, but this case is worse. Children who witness abuse to their parents struggle significantly with it. Any abuse, even what some may call 'minor' (which it is not!) such as forcing you to kiss him, does harm to the child. It teaches them to be afraid, and that their parent can't be trusted to care for them, and as a women it teachers her to expect this sort of abuse which is can lead to her ending up in an abusive relationship when she gets older. This is just touching on some of the harm that it does, this is something you do not want your child to witness!
Furthermore, he is forceful and abusive, and that abuse is not likely to be limited to you. It's just a matter of time until he hurt your daughter if he is allowed near her, and I think it's obvious how this can be harmful. It's much much better for your child if she never has to know him!
I would also mention that I've also advised women in abusive relationships, not nearly as often but I have done it. One thing I have noticed is that the longer one experiences abuse the more willing they are to tolerate it in the future, and worse the more they feel they can't do anything about it or even convince themselves they deserve it or that they can change the man somehow. For this reason I strongly advise you to cut him off entirely for your sake (which also would be for your daughter's sake). I'm glad you saw the abuse and left him, that shows your strong. However, keeping away from him prevents him from hurting you and helps you to feel strong enough to be able to resist any attempt of him, or other's, to take advantage of you in the future.
You mentioned the father's 'right' to contact to the child, so I will also address this. He has no right if it harms the child! In all cases the child takes priority over everything else! The moment that a child is harmed by it that's child's right to a healthy upbrining trumps any other theoretical right a parent may have! His abuse and mistreatment of you have caused him to give up any theoretical right he may have had. Besides, as I know men like him it's unlikely that he cares that much about contact with the child and he won't suffer from lack of it, he most likely uses contact as an excuse to visit and try to pressure you.
So in short, don't let him have contact! even if he says he has changed don't do it. People rarely change, and men like him will often claim to have 'changed' to worm their way back into someone's life only to then go back to their same habits. Just stay away from him however you can, both you and your daughter will be much better off that way. You both deserve better.
