I know the OP ended up choosing the GSXS, but for others that might be researching, I thought I'd give my quick thoughts also.
I bought my FZ-09 in 2014, first model year. Was the throttle twitchy, yes. Was the rear suspension a pogo stick, yes. Was it a hoot to ride? **** yes! The motor is a gem and it is a budget bike. All those things are addressable. Now, compare that to the GSX S-1000... Well, out of the box, the GSXS beats it soundly everywhere except for low end torque. The GSXS is more planted, more stable in corners.Better brakes. The acceleration once the revs are up is not even in the same ball park. The GSXS is on the "next" level. Both bikes have had some updates since their original models, most of their first gen problems have been addressed. Unless you are only interested in driving around on one wheel all the time, and you can afford either bike, I believe the GSX S-1000 is the better bike for most people. But, like always, test ride both if possible, to see for your self.
2013 Honda F6B