look for a lip balm with kojic acid. Kojic Acid is highly effective for hyperpigmentation, especially on the lips. I would also suggest ALWAYS protecting your lips with an SPF balm, to prevent further damage from the sun.
Also, gently rub your lips at night and in the am with a oil.
You could also use a serum with kojic Acid (yes, you can apply it to your lips). just be sure it only has other hydrating ingredients with it; such as Glycerin, Hyaluronic Acid, Shea Butter, oils, Niacinamide, Squalene, etc; (just be sure nothing irritating is on the ingredient list).
There is a store front on Etsy that makes a kojic acid balm that is highly reviewed!
The store is called; BOTANICAL DELIGHT. (pink lips lightening balm)
It's PACKED with moisturizing ingredients & Kojic Acid!