The ninja 300 is a nice bike regardless, but there is no way in hell it will be any heavier than a 600cc. I promise you that you will get bored of it quick. As soon as you ride along side a 600cc while you're on your 300cc, the struggle to get to the 600cc speed sucks! So yes, you will outgrow it fast...once you reach the max throttle and such, you will want to move faster, trust me.
I road a my buddy's 250cc the other day and I couldn't take it. The only thing fun about a 250-300cc is you'll be able to throw that shit around! But with time, you'll be surprised that it's very possible to do that with a 600cc. If it makes it better, I am 5'6" and I have no issues riding my 600cc.
Once the 600cc is in motion, it stands on its own and you can maneuver it quite easily. It's just a matter of how much you move off your center of gravity when you're leaning and turning.
Shit, Now I want to ride! Spare yourself and go straight to a 600cc.