Hmm, I take it you've tried contacting your clinic for advice? (Maybe they'd give you a cgm for a fortnight just to sort it all out?)
Could you blood test when you child(ren) wake to give some night time readings? (Just want to rule out a night time hypo.) There's also the dreaded dawn phenomena (sign), which is another possible reason for a breakfast raise.
There's also the possibility that your lantus isn't quite lasting 24 hours... (You could swap to a night time dose...)
Or if you did a mid night read you could skip/delay breakfast to see what's happening?
Hopefully others will make some more suggestions. Good luck, I remember those desperate sleep deprived days of yound children and am just grateful that mine are now in their 20s.