I had peeling lips among my other skin problems and I found something that really helped.
First every day I take one Cod Liver Oil capsule.
Also I take probiotics.
But the thing that has helped the most not just with my peeling lips but my skin in general, is Elomite Cream. It is to kill Demodex Folliculorum. It has to be prescribed by a doctor. In this case my dermatologist prescribed it to me because nothing else we tried was making my face itch less. Since I have started using it I have noticed much better skin and my lips do not peel at all anymore. They used to peel incredibly bad. It used to be that I could pull off almost all of the skin on my lips every 3 or so days. Now My lips do not peel at all, not even tiny flakes. But the Elomite Cream is not supposed to get in your mouth so I don't think it is supposed to be used on your lips, but I use it on only the parts of my lips that can be touched with my mouth closed.
I am not sure that this will help any of you, especially because it will be hard to convince your doctor to check for demodex, much less treat it. My dermatologist has been a professor for many years and actually has been research doctor for many years and so he has lots of knowledge of many skin conditions, and thus had the idea to treat for demodex.
I hope I helped somebody, and good luck with your lips.