What are the ages of the children? If they are over 3 enforce a simple rule that they do not leave the interior of the boat without a life jacket on. If younger and mobile it gets more difficult. How many adults will be on the boat? Will there be alcohol use by the adults? These are all factors in what I would be comfortable with in regards to the children. I don’t think the netting provides much added security. You could certainly purchase netting and bring it down with you. Plan on a couple hours to rig it but kids can easily circumvent it. Keep the adults sober and the kids in life jackets. Anytime you are underway keep the kids in the cockpit. If they want to go out of the cockpit an adult has to go with and remain with them.
Most important is to bring life jackets with you for each child that they are comfortable wearing. Don’t rely on the charter company for the jackets.