I suggest a skin doctor (dermie?) Toothpaste is often blamed for this kind of thing, most often inaccurately. Try 1% hydrocortisone cream a coupole of times a day while waiting to see the doctor. Your response (or lack of response) may help clarify the situation. I think you may have cheilitis, or eczema of the lip. This usually has no specific cause, but is easy to treat.
I can't think of any connection with rosacea.
Best.
Dr. Rockoff