It's a 2009 stock suspensions, my weight is 175lbs (at least 180lbs with gear?).
Another way to describe this: the rear feels "light on the pavement" causing some instability when entering corners. It's subtle, and I learnt to live with it, but definitely noticeable compared to the other bikes.
At first I thought the Michelin Power Pilot weren't as good as the Q3 I had on the other bikes, but then after switching to Q3 on this bike I noticed the benefit is more apparent at full lean angle and accelerating out of the corners, rather than entering.
OK, that's some info.
Where is your preload set on the back? The 09-12 CBR's come with way stiff springs (1.0's if memory serves me) stock that work great for us fat guys like me but are too hard for the normal guys like you; if you are running high preload on the rear shock (you should probably be at the stock position or one lower) and/or low preload (compression settings would also make a difference here) you could be underlapping the rear and overloading the front making it feel twitchy. Springs in the .90 or .95 would probably suit your weight better. It's not a definite thing, though, people feel things differently.
PP's are good tires, you shouldn't have much difference between them and the Q3's, though I find PP's to have a weird tip in where it sort of falls to tips into the first part of the turn in then hook up and feel confident and stable, Q3's should be more even all around (in full disclosure i've not ridden a bike with Q3's though Dunlop's usually are even feeling). Someone with more experience should be able to chime in.
What tire sizes and what pressures do you run? That can be a factor as well.
When was the last time the suspension was serviced or setup for you and your weight?
Do the forks stick out the top of the triple clamps at all? Can you take a pic if they do?
Again, the bike should be stable, I think you might be chasing setup issues.