If it is clogged (bacteria) glands causing your EC, you could try a blue light. It is a light therapy product that is used to kill bacteria in pimples on face. It is harmless to skin (you can research to see) unlike some other phototherapy products. Although a topical ointment might not be able to get into a clogged gland, a blue light can. Some dermatologists have this blue light, but you can also buy it for home use (I think the price range is $100-$200). It seems like if this is the same kind of oil that causes pimples on face, then what works for the bacteria deeper down in skin for face will also work for the lips.
Oh I just saw this treatment listed with blue light. I have not looked further into it. I copied and pasted it:
Pulsed light and heat energy (LHE) therapy. This treatment combines pulses of light and heat, which researchers believe target two causes of acne. It is known that LHE destroys P. acnes, the acne-causing bacteria. LHE also may decrease sebum (oily substance) production by shrinking the sebaceous glands. A system that combines pulses of green light and heat has been approved by the FDA for treating mild to moderate acne.