Could a genioplasty advance my lower lip? (Photo)

Hello dear
Thanks for the question and provided information as well. 
My recommendation for you is a personal consultation with a certified plastic surgeon to be able to make a complete evaluation and suggest the procedures that can fulfill your expectations and provide you with the best results.
Good luck :)

Hello,
The teeth are what support the position of the lips, not the chin. A genioplasty should not be used to correct a malocclusion (abnormal position of the teeth) where your bite is not in the best occlusion. Genioplasty can be combined with a jaw advancement if necessary. Your lower lip labiomental sulcus will likely deepen if you do have a genioplasty advancement but will not improve the lip position. Without further physicial exam, it can be difficult to determine exactly what you need, and would recommend you seek an experienced maxillofacial or craniofaical surgeon in your area. 

If malocclusion exists, then jaw advancement is indicated. This movement will open up the labiomental fold and give height to lower facial 1/3. 

It is important to recognize that the labiomental sulcus is a fixed anatomic point of the chin. It will not become less deep from a sliding genioplasty (as it lies above the level of the bone movement) and can even get deeper as the chin bone moves forward. Filling in the bony stepoff of the sliding genioplasty can help lower the risk of labiomental fold deepening. But it certainly will not make it become less than before surgery. Making the labiomental fold less deep, whether a sliding genioplasty is performed or not, requires a direct approach to it.

Any chin alteration will not move the lip forward and could actually make the lip more upright. The lip gets support from the teeth so your best option to balance the upper and lower lips is a lower jaw advancement. So correcting the bite and the profile  at the same time