Can a parent challenge a temporary care and custody agreement in family court instead of child protection court?

Generally, no - a temporary care and custody agreement is a matter for the child protection process, and challenging it directly, for example on the basis it wasn't properly consented to, is something to raise with the society itself or, if it becomes a formal court matter, the child protection court, not the parents' own family court case. Family court's jurisdiction is over parenting time, decision-making responsibility, and support between the parents - it doesn't have authority to unwind an agreement a parent made with a society.

Where the two can intersect is that the existence and terms of a temporary care and custody agreement are relevant facts a family court will want to know about when deciding a parenting case, and a parent can certainly argue, within the family case, about how the child's current situation should affect parenting time or decision-making going forward. But if the real complaint is about the agreement itself - that it was signed under pressure, or that the society isn't honouring its terms - that's a conversation to have directly with the society, or a matter to bring before the child protection court, with a lawyer's help to identify the right forum.