Hi, H.. My middle one had a strange and similar situation as yours and, like yours, she was around 2 years of age. It was a rosy dry patch that surfaced more prominently on the very dry winter days and it was just under her lip and a bit around her chin. It was like pink sand paper, very strange.
Eucerin's Aquaphor did "ultimately" work the best, just be patient. But I had to keep it on hand and reapply it throughout the day - but not unlike how often we put Chapstick on our lips, if you think about it. She's almost 11 now and still gets a little "something" after she's been playing outside and its super cold and dry.
Remember, too - growth spurts and the hormone surges that accompany them do induce these skin eruptions and changes. That could be adding to this. I noticed a pattern with mine in this respect. Good luck!