An acceleration sensitive fault? Weird. Can you get it to stutter in 3rd if you're accelerating hard up a steep hill?
Does it backfire when it exhibits the fault? (Which would suggest sparks getting killed and unburnt fuel getting into the exhaust.)
If not, it points to fuelling... perhaps float heights are wrong, with a low level of fuel slopping to the back of the float chambers under acceleration you could be running into fuel starvation that doesn't show up under the lower acceleration you get in higher gears.
Most electrical problems are dodgy connections that would probably show up with bumps in the road rather than acceleration... the only possibility I can think of there is the tilt sensor, which IS designed to cut the ignition if the bike's on its side. Perhaps if that was misaligned or damaged in some way it might give the symptoms you describe.
Edit: Strike that last bit. No tilt sensor on the FZS600. :o