If we have to take JR action in our child’s name and lose, could we or our child be ordered to pay the other side’s costs?

Assuming this is with the benefit of legal aid, almost certainly not. The law takes a pragmatic view that it is pointless to order someone who has no money to pay costs. There might be an order for costs which is not to be enforced without the leave of the court – that allows for the situation where the child suddenly comes into money somehow. However, neither we nor any solicitor we deal with regularly have never known any such order to be enforced.