You probably can take him to either court. I am assuming that you have a child support order which directs that he pay half of any medical bills for your child. Small claims court would be able to enter a judgement after a hearing that he owes you half of the bill. You would need to bring in a copy of the order. I would recommend a certified copy of the order, you get a certified copy from your Family Court, or the court that issued the order. If you take it to small claims court you would only get a judgment.
It would therefore be better for you to bring a violation petition in Family Court. I am assuming that you have an order of child support which includes a provision that he splits medical costs 50:50. His non-payment, if it is so ordered, is a violation of the order. In most states a violation would subject him to penalties, such as jail time for non-payment (Judge is not likely to incarcerate anyone on the first violation, but if he makes a practice of not paying then the Judge may decide to jail him eventually). More importantly, if your case is going through your child support enforcement unit, once the amount he owes you is reduced to a specific number they can assist you in collecting the money. They are unable to collect it for you when your order says 50/50 because they do not know what amount to collect, but as soon as there is an order saying a specific dollar amount, they can add it to their collections. This might make it easier to collect from him.