Hi. I had splits in my lips at the corners of my mouth and they were so sore. Yours may be a different problem ? Anyway a friend told me it was a shortage of good oil in my body and suggested cold pressed olive oil. I buy it, use it and it works - sometimes overnight when i feel it coming on and other times after a few days.
I'm an adult and initially took several tablespoons a day so you would have to work out how much to use for a child, which would be considerably less and maybe spread out over the day.
I took it all at once shortly before retiring to work its magic while i sleep. Not very nice but we do what we have to.
Now i try and add olive oil to salads and other food as a preventive measure and my lips have been fine for several years now.
Worth a try and cant do any harm.
Also its available everywhere, cost effective and natural!
Best to you and your 3 yr old, fitter44