So the left STI lever is catching when shifting from the small chain ring to the big chain ring? Does it squeak or make any other noise? Have you tried removing the hood and spraying the mechanism with a lubricant (e.g.,:
Amazon.com: DuPont Non-Stick Dry-Film Lubricant Aerosol, 10 oz : Tools & Home Improvement)? Short of that the apparent consensus is that the mechanisms in STI levers are not user-serviceable.