In regards to telling the court about why the child was withheld, all of that needs to be included in an affidavit and filed with an initiating application. While there are no parenting orders in place, the child can live with whoever and the parents are at liberty to behave as they wish.
What will come into question is whether or not you've acted in the child's best interests by separating her from her primary attachment figure on a permanent basis. As such, you need to show why your actions were in the child's best interests. What concerns did you have about the child living with the mother? Was the child at risk of harm?
In relation to the psychologist, this is also another tricky situation because the court doesn't generally take to parents who unilaterally conduct major medical interventions without consulting the other parent. It may not make or break things, but not involving the other parent at all suggests the father lacks insight as to the mother's role in the child's life. I'd also be inclined to wonder if any distress the child is experiencing is perhaps linked to the separation from her mother. Her mother may be an unpleasant individual, but if the child predominantly lived with her, then it's likely the child's primary attachment figure was her, and studies have shown that separation from a primary attachment figure can cause significant anxiety in young kids.
Fundamentally, yes, telling the mother about the psychologist and the psychologist's contact information may lead her to become hostile, but it will show you recognise she is an important figure in the child's life, that you value her inclusion, and that you're willing to co-operate. I can't stress how important the notion of 'friendly parent' is in this day and age. If one parent is seen to be unwilling to support the child's relationship with the other parent, or feels the other parent is not important, it can, and frequently has, cost them heavily in court.
I also think it's important to remember that you don't have to enter every argument that you're invited into. Send an e-mail, inform her of the situation, and ignore all hostile responses. Don't enter into a back-and-forth argument about the situation - it just acts as fuel for the fire.
And most importantly, I think it is very important that you try and arrange the circumstances so that the mother spends time with the child, even if it is supervised. I also think a post-separation parenting course is very important. It'll teach you how to communicate without escalating hostility.