Flattened Lips After Botox

Botox relaxes muscles. It does not fill in wrinkles. When Botox is used on the lips, the orbicularis muscle (the circular muscle that surrounds the mouth) relaxes in areas where the Botox was injected. This reduction in muscle activity can result in a flat appearance or reduction in function of the orbicularis muscle.
Rather than injecting the opposite side and ending up drooling or speaking funny, you may want to let the Botox go away, then use a hyaluronic acid filler like Juvederm to reduce the lip wrinkle.
Also an option is laser resurfacing if the lip lines are due to sun damage.
Last, smoking and sun contribute greatly to the formation of lip lines, so avoiding these will minimize the development or worsening of lip lines.

You have to be careful with this. Most fillers will last beyond the time of the Botox and then you will have caused a different problem. I would generally advise waiting until the botox wears off. IF you are intent on using a filler, I would consider a short term filler such as Prevelle silk.

It may be best to wait until the effects of the Botox injection wear off. Injection of Botox to the other side of the lower lip would also flatten out that side - probably not the symmetry you would be looking for. You may want to find out how many units were injected to that wrinkle for reference. Filler injection may be an option to help correct that vertical wrinkle in the lip in the future.
Good luck.