Your eyes appear prominent & hence look "big". Trying to hitch the lower eyelid up, when you have prominent eyes, is akin to tightening a belt over a beer belly- it just makes the protrusion more obvious, thus I would not advocate a lateral canthopexy.
Although a higher risk, lateral orbital decompression, will create more space in the eye socket & allow your eyeball to sit in a deeper position, thus not appearing prominent. It will allow the lower lids to sit higher too.
To a certain extent this can be achieved by a lower lid spacer - but the aesthetic results maybe modest & short lived.
A lateral wall decompression is a common operation to treat prominent eyes due to thyroid eye disease, but, there may be very few surgeons willing to do this for a cosmetic indication, as any decompression surgery has a small risk of sight loss.
A canthopexy is procedure used to elevate the lateral corner of the eyelids and correct lower lid lag. This may be the procedure that would address your concern.
There certainly are surgery that can be used to alter the shape of the eyelids. However, without a personal consultation it is very difficult to know if any of these procedures will be right for you. I would proceed cautiously with this project. It is one thing to promise a result and entirely something else to actually deliver the result. I think it is important to consider the over all shape of the face and how this affect you thinking about the eyes.