New Zealand Law
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Under the family court system he cannot just file an application for increased contact time but must start with a mediation with a mediator appointed from the lists of trained family mediators. This resolves 80% of cases. But if it doesn't work, then it is possible that he could get more time but the court would appoint a lawyer for your daughter and there may also be a need for psychological reports. The real issue is what is in her best interests. If he is just doing this to score points then he is unlikely to succeed. The time needs to be managed carefully when the child is so young and any change should be gradual . He must show a definate plan for what he does in the increased tome as it cannot be just random.
Sadly I have seen similar cases, and doctors as narcissists seem to feature. But a gradual approach is best rather than any sudden change and that is how the courts would approach this. If he says he has a strong case then he is just taking a position to intimidate you. In reality the court won't move quickly to change what has been operating satisfactorily. If you intend increasing his time with her gradually and make sure this is on the record, then you will be seen as reasonable. If he postures and demands the court will be cautious. And if your daughter is reluctant then the lawyer appointed for her will advocate for her and explain this. The wait time for a mediation can be just a few weeks. Then if a court case is needed you would wait at least 6 months and possibly longer
Glad to help