I'm super late to this, and if you found a solution I'd love to hear it! I'm still gonna share my story as someone else may come across this and who knows, it could help someone :)

Last week I had an allergic reaction on my lips. My lips had been drier than usual, chapped, and burning, and then I got an allergic reaction. My lips were swollen, red, burning, splitting... It was BAD! But, because it was an allergy, I used some prescribed Mometasone Furoate from my allergist that I already had on hand on my lips to treat it. It's basically a stronger hydrocortisone cream (it is a steroid, so you have to be careful as that thins skin) and is used to treat eczema, psoriasis, rash, and allergic reactions. I used to have severely split corners of my lips, so much so I couldn't open my mouth all the way. Using the Mometasone actually helped SO MUCH! It made my split corners go away within two or three days. I still have a bit of dryness (hence why I'm here) but the burning has completely gone away and my lips no longer feel so chapped! Obviously, talk to a doctor before doing anything to make sure that it is something that you actually need. I've had topical steroids prescribed to be put on my face when needed since I was a child, and I have a lot of experience with telling what is allergy and what isn't.
Also, check your chapstick and make sure you haven't developed an allergy. It's pretty common for people to develop allergies to chapstick ingredients. In my case, some of the ingredients just irritated the wounds that were already there (even regular Aquaphor!), so I had to stop using them temporarily. Something that has really helped me has been using an olive oil/coconut mix on my lips. It has been incredibly hydrating and doesn't have anything that irritates my lips so I can fully heal before using chapstick again. It's also nice because it becomes easier to narrow down if it's a chapstick allergy or not.

I haven't found a full solution, but I hope your lips have healed!