Can a botched labia reduction procedure cause urine to leak ?

Not likely: Labia reduction surgeries don't generally affect the urinary tract. The surgery is confined to the labia and sometimes the clitoral hood. But this is well away from the urinary tract. It is doubtful that a labial reduction surgery would cause urine to leak. Best wishes.

Unlikely: Labia reduction, either minor or major, is far enough away from the urethral orifice that it is highly unlikely for surgery to cause urinary incontinence. However vaginal rejuvenation is significantly closer to the urethra and sphincter. This type of surgery is more likely to potentially cause incontinence.