If your children won't eat off of the menu (there's a burger, wings, short rib sliders, etc. could be modified to be less spicy or whatever they'd need), I'd feed them before you go. Not sure they offer kid food there.
When we dined in the lounge our server informed us that the entire restaurant menu was available. So, your child could get the prix-fixe child's meal from the restaurant.
When I was seated in the lounge section near families with kids, they were just sharing apps from the regular Lounge menu. But as PP stated, you do have the option of getting the prix fixe meals when seated in the Lounge.