I do genuinely find it odd that I have had so much engagement on mumsnet with mothers who have been victims of violence but say the courts don't take it seriously - but all my experience in court has been the exact opposite.
If a woman had a split lip from a partner, I would be astonished if a judge did not take that very seriously. Violence from one parent to another is seen as a serious failure of parenting, even if the violence wasn't directed at the child the child may have heard it, or seen the aftermath. This is incredibly emotionally damaging and judges know this full well.
All I can assume is happening is that the split lip or similar incident happened several years ago, the man denies it and there is no evidence other than the woman's word, which may not be enough, particularly if she didn't make any complaint at the time.
So I guess the best advice must be - complain, document, tell someone as soon as you can. The court can only do its best with the evidence in front of it.
And I am afraid i have many examples of women exaggerating or making up serious allegations to thwart contact. Just as I have many examples of violent disgusting men who shouldn't be allowed near any child.
Neither sex has the monopoly on appalling and stupid behaviour when it comes to parenting. But in this case you have findings of fact by a court that have not been challenged, so I am afraid you are stuck with them - unless you want to abandon entirely the rule of law of course.