Yeast or candida is a good suspect. There is a prescription mouthwash that works great. Nystatin, oral medication is often prescribed for thrush, an overgrowth of candida on the tongue and ora mucosa. If it's attacking the lips, its all over the mouth too. Lips like toes are always almost wet or almost dry and both make wonderful homes for skin yeast infections.
If by chance it doesn't respond to just nystatin the infection may be chronic herpes simplex. Usually fever sores show in the corners or cleft of the lip, split right across at the middle of one or both lips. Zovirax is the only effective antiviral drug and it comes as ointment, or in pill/IV. If Abriva helps, it may be to the drying action of the medicine although, it is a top OTC fever sore ointment. This could point further at herpes simplex that over 80% of people harbor. Zovirax ointment will be soothing and a viral infection like this will substantially improve within 7 days. The zinc and such may be a cure, the supplement L-Lysine is a must for anyone fighting viral infections, and it is a well tested adjunct medicine taken with any form of Zovirax.
Excepting the potential of having both fungal and viral infection together, don't count out a staph infection, maybe a lousy S.a.. All sorts of antimicrobial ointments like Bacitracin can treat the most resistant bacteria. This will tell you if the infection is bacterial. A recent inability to breathe due to a head infection can dry the mouth and lips especially during sleep. Good old petroleum jelly to the rescue. I can't stand it anywhere except between me and my bicycle seat so I smear coconut oil on my dry lips.
Honestly, don't allow split lips to go untreated. Neglecting a skin split often ends in surgery, not the most enjoyable experience on or in the lips and mouth. I had surgery on my lip corners twice as a kid. Apparently, I used to drool in my sleep.
Here's hoping you find an answer soon.
LL