OK so the recommended fallback among some folks in the know, if you can't land tickets on the Coopculture site, is to get in line at the NEW onsite ticket office before it opens in the morning and buy your tickets there. You'll get your kid's tickets at the same time. I am reading that your entire group needs to be together and ready to start your Colosseum visit at the time of purchase. You can't, say, buy them for the next day. You'll most likely be assigned a time slot that's for nearly immediate entry. Bring your kids' passports for proof of age; probably most important for your 16 year-old but you're always supposed to have them on you anyway so....
These will most likely also be general-entry tickets but it's possible that you could land a tour that includes the arena. I'm seeing that they do not sell tickets for tours of the underground. Honestly, this could be a lot less painful (and even more successful) than trying to land tickets on the Coopculture site.
https://colosseo.it/en/2023/05/ticket-office-opening-colosseum-square/