Look, I’ve been down this rabbit hole myself. The short answer is: you can’t see their archived posts from another account. Instagram keeps that folder locked to the logged-in user only.
Here’s the thing though - if you’ve got monitoring software installed on their phone (which is the whole point of being on this forum), most decent apps will capture screenshots or log activity before they archive anything. That’s been my saving grace more than once.
The real question is whether your monitoring app is actually capturing their Instagram activity properly. Some work better than others. What are you currently using? Because if you’re relying on trying to crack Instagram’s privacy features instead of having proper monitoring in place, you’re making this harder than it needs to be.
Also - and I say this as someone who learned the hard way - kids archive stuff specifically because they know parents are watching their main feed. The fact that you noticed this behavior is actually the important part.