I'd look into getting a dedicated track bike. If your gonna start doing more track days, your gonna get the bug and want to upgrade to a better bike anyway. The instructor was right, the FZ6 is not a track bike. It's still fun, but not a race bike.
Also consider the fact that, if your getting into track days, your going to be pushing your limits more and more, and your very likely to crash. It's a lot cheaper to replace the race fairings, and get new/used parts for the superbikes than it is for the FZ6.
A dedicated track bike can be had for pretty cheap too. If I were you, I'd find some of the local racing clubs/associations in your area, and see if they have an online classifieds section. I've seen some awesome deals on the
WMRRA classifieds. You should look around just to get an idea of what it might actually cost for a track bike.