First bike took me like 30 minutes of trying, and my time was 2:00 or so. Second time was a week later, 2 times, 2:01; third time was immediately after and took a million tries, 1:46.512; and I got tired of racing that track so I don't have the 4th yet.
The key for me riding that bike is to brake late and turn early, going deep in the corner, then grab all the throttle you've got and get the straightest run at the straightaways as possible. Doing this will allow you to pass the first guy usually before the right hander to the Dunlop bridge, if not well before - if you don't make it by then, be sure to get on his outside, or underneath him after the bridge. He rides like an a**hole, and pays no attention to where you are. If you cannot pass him before the big hairpin before the long straight, start over.
The second guy isn't much faster than the first guy, just farther away. Again, outbrake him and out accelerate him by making the straights longer. You need to be within 2 to 3 seconds of him after the first lap to have any chance of catching up.
You can make up tons of time on them by braking late and turning early - this will put your apex well outside of the inside of the corner, but it'll also give you a sharp and long straightaway; I found my exit speeds to be several mph faster by doing this. As you're flipping around, gas it earlier than you would think - there's really no wheelspin in this game, and you'll be going quite slow at the apex anyway, so gas it and gas it hard!
This is the only bike that really responds well to riding it like that. I tried on the other two strokes, and even the 125, and it just slows you down. Being smooth, however, doesn't seem to appeal to the 250 - you just have to ride hard and quick.