I'd post in legal and ask for advice from someone legally qualified to give it, but as a general rule of thumb: courts like the status quo kept for children, pretty much at all costs unless they are actively being harmed by their living situation. If you are primary carer, always have been, and she is strongly bonded to you, then he hasn't a hope in hell of winning sole or even primary residence unless you're actively abusive in some way. Children are seen as better off where they are used to being, with generous contact with the other parent. In most situations, anyway - family law is notoriously individualised.
Having said all that, if his ex denied contact then this may open wounds, and I'd be careful to propose generous contact, with a schedule leading up to overnights when she's a little older if he and you split a while ago, and earlier if the split is recent and he and she have a strong bond. It will also be really important for her that she and her dad have a good relationship.
Is there any way you would think about going to mediation to see if you can get a court order by agreement? You could stress that you aren't trying to exclude him and want him to play a big role in her life, acknowledge how horrible missing out on some of his other kids' lives has been, blah blah blah? I know some people are impossible to work with, but if you can try to get it across that you need to work in a polite and cooperative way to do what is best for her, rather than making her a battleground over which you fight, things might go better. If he loves her, then he may take that on board (and certainly your taking it on board will go a lot better if it ever should reach a court hearing).
I'm not for a second denying that he sounds an arse, I'm just saying that if he does love her, and you can just get it across to him that you need to work as parents, even if apart, to give her a calm and stable childhood, then that will be massively easier for you and massively less harmful to her. Parental conflict is so bad for kids. He needs to stop seeing her as a possession to be won and to start seeing her as as a small person whose needs must be met as well as possible by her parents, regardless of what makes them happy personally.
There is one other point. A friend of mine is a family barrister and he says most of the people he sees want the judge to vindicate them - to "win" and have their version of the marriage and the other's parenting ability stamped as the official truth. What usually happens is that a middle course is struck, and both sides feel angry, bitter and that the other side conned the judge/CAFCASS. If you can get an agreement with him in mediation, and that formally stamped as court-approved, that would be cheaper, gentler and a lot more likely to result in good things for you and your daughter.