Welcome to HealthcareMagic forum.
Itchy and sweaty feet in children with peeling skin might be due to a fungal infection
which is common in moist intertriginous ( between toes ) areas of feet .
The best way to get it treated is to consult a dermatologist
who can confirm the diagnosis by thorough evaluation and accordingly prescribe antifungal ointments and powders containing ketoconazole
or terbinafine .
For itching & peeling ,lotions containing zinc oxide ( like calamine etc) might help .
You are right in thinking that too moist and always covered feet may worsen the problem so keeping the area dry with antifungal dusting powders is recomended .
If itching is not controlled with local applicants then oral antihistamine like chlorpheniramine
or hydroxyzine might be prescribed by your doctor .
Hope this answer is helpful to you .
Wishing your child good health.