The courts presumption is that when children live with one parent they (the children) have the right to see and know the other parent in all but the most exceptional cases. Children who are insecure about their parentage and heritage tend to grow up with low self-esteem leading to behavioural problems, emotional ill health, dysfunctional relationships in adulthood etc etc. Rather than opposing contact altogether it is often better to concentrate on securing safe contact.
The principles set by the Court of Appeal as law relating to DV and contact can be summarised thus:
i) the Court should consider the conduct of both parties towards each other and towards the child, the effect of the violence upon the child and on the residential parent, and the motivation for the parent seeking contact, ie is it a desire to promote the best interests of the child or a means by which to continue violence, intimidation or harassment of the resident parent;
ii) on an application for interim contact, when the allegations of domestic violence await adjudication the Court should give particular consideration to the likely risk of harm (physical or emotional) if contact were granted or refused (any risk of harm to the child must be minimised and the safety of the resident parent as well as the child should be secured before, during and after any such contact).
iii) There was not, and should not be a presumption that on proof of domestic violence the offending parent had to surmount a prima facie barrier of no contact. As a matter of principle, domestic violence of itself cannot constitute a bar to contact but is one factor in the difficult and delicate balancing exercise of discretion to be undertaken by the Court.
iv) In cases of proved domestic violence the Court has to weigh the seriousness of the domestic violence, the risks involved and the impact on the child against the positive factors. The ability of the offending parent to recognise his/her past conduct, to be aware of the need to change and to make genuine efforts to do so would be likely to be an important consideration when performing that balancing exercise.
v) The rights of the child must prevail.
There is usually little point complaining about CAFCASS. If there are errors of fact or law, the place to raise these is in the subsequent directions hearing (or final hearing as appropriate). A lawyer will handle this for you.
Evidence of incidents and/or the effects of DV on you and the children in the form of police reports, GP letters, school letters etc are in themselves not usually enough on their own to persuade a court that contact is not in a child's best interests. They can be used to argue for a finding of fact hearing and independent expert reports.
I'm not sure what it is you are wishing to appeal at this stage. It's not possible to appeal a court decision just because you do not agree with it. It needs to be shown that the judge has made an errors of fact or law, there are material facts which have come to light which weren't known at the time of the judgment or the ruling flies in the face of what would normally be expected.
Contact orders can ultimately be enforced with community service, imprisonment or changing residence so that children live with the other parent.