What kind of treatment will permanently get rid of 'pebbly' or cobblestone chin?

The cobblestoned appearance of your chin is due to the action of the underlying muscle (mentalis). Botox is an effective way to relax this muscle and prevent it from creating that "pebbly" look.. 5-10 units are usually sufficient to produce a nice result. It is not, however, permanent, and results will typically last ~3 months.

The pebbled appearance of the chin can be improved with 10 units of Botox. Over time, the pebbled appearance may become less noticeable and less Botox may be needed.

Botox can work well for cobblestone chin. It does not take many units and it's only one or two injections. The effects are temporary and would need to be repeated, but it's a quick and straightforward treatment in my hands. Can't say for sure without seeing it in person. Best of luck.