I'm sure you've moved past this by now, but I've had some success with stripped screws using a comparable, if slightly larger flathead screwdriver. The screwdrivers that come in those kits tend to suck quite a lot... iFixit actually has a guide for stripped screws: Cómo quitar un tornillo barrido

Not having the isolation pick, I used 2 thin guitar picks instead, which did the job fine.