It's in the child's best interests to get the agreement finalized in court. It's also in the parents' best interests. Without a court order, there is no legal obligation to pay, and the CP would end up back in court anyway. Also, without the order, anything the NCP pays can be considered a gift, and if the CP files for support later, then it's possible it can be ordered retroactively (depends on the state).
Just get your agreement ordered for everyone's peace of mind.