What's the best way to deal with a stubborn 4 year olds chapped lips?

Emollient. There are a number of excellent emollients on the market. Simple vaseline is cheap and very effective. You may want to consider using vaseline prior to the child going to bed. If he is resistant, wait until he is asleep and apply it. You can also consider using a flavored balm. The taste may help with compliance.
Hydrocortisone 1% Persistent chapped lips can be lip eczema, which may be treated like skin eczema, using Hydrocortisone cream (twice a day for 1 week) plus vaseline or aquaphor to retain moisture. Sometimes lip balm or lip sunscreen chemicals cause an allergic reaction. Other times, a person is a primary lip-licker (it's his habit). Once cracked, lips get irritated by routine foods and juices. A doctor can help.