Willl a Corner of the Mouth Lift Correct my Asymmetrical Mouth?

There are several possible things that contribute to the shape of the smile, and so different options depending upon the specific condition.
One possibility is that the bone structure of the face is a little bit different, as is the case for example with Brian Williams of NBC News. Often it is related to muscle activity and in that case, some Botox to the higher side can work, assuming you don't mind lowering the higher side.
If you do decide on a lip lift, thre are a couple of ways to do it, and one that I like places the scar along the bottom of the nose where it is very inconspicuous. But if you have ever seen the mirror image pictures, where one side of the face is reversed and matched up so the face is perfectly symmetrical, it usually looks kind of odd.

There are multiple anatomic abnormalities that can result in
asymmetry of the mouth. These may include abnormalities of the underlying
facial bones, teeth, and lip soft tissue. In most cases asymmetry of the mouth is relatively minor, but in some
cases may be severe and extremely noticeable. Depending on the cause, treatment
may or may not be possible.It’s hard to tell exactly what’s
going on from your pictures, a physical examination that looked specifically at
occlusion, teeth and the facial skeleton would be helpful. Asymmetry of the
facial bones can be a major contributor to this type of problem. Dental
abnormalities may also contribute to this problem.It’s therefore important to localize
the cause of the problem. Depending on
the cause a wide variety of treatment options may exist. The results of
treatment are dependent upon applying the right treatment to the right problem.
When the problem is related to the
soft tissue, lifting the corner of the mouth may be associated with high
satisfaction rates. If you’re considering this approach, consultation with a
board certified plastic surgeon who has experience in this area is appropriate.

No. The problem is a static and dynamic issue with regard to the upper and lower lip. Minute injections of Botox into the strong side might help alleviate some of the asymmetry, but this is not a good idea to have surgery on this dynamic asymmetry.