Welcome to the best Kawasaki technical support site on the web. Many of the posts for the KX500 also apply to the KX250, so you will no doubt find a lot of useful information for your scooter.
Enhancing the turning of your KX250 is really quite simple, as turning is a manefestation of steering angle. To enhance the turning of the 250, raise the forks in the tripple clamps a little at a time until you have the turning characteristics you desire. Keep in mind the more the forks are raised above the top tripple clamp the more unstable the bike will become, resulting in head shake. I would expect that you will find that by raising the fork above the top tripple clamp between 1/4" and 1/2" will give you the turning ability you desire. My KX500 is set at 5/16" above.
You may also want to address gearing, as this will enhance the power characteristics of the 250 to allow you to use power in cornering. When riding a 125, you don't have much power on tap to aid in turning. As the CC's increase, power become a bigger part of turning. On the 500, power is a huge part of getting around corners at speed, and one must learn to drift the rear wheel in corners to maintian speeds while holding a consistent line.
As an example, last Sunday I rode with a group of six guys, riding almost exclusively in the washes & draws, including tight technical draws. Bike sizes ranged from 125-250-426, and of course, my KX500. Since I am slightly competitve, each time we stopped to gather everyone in a group, I would let everyone else start in front of me. With my bike setup and the awesome power of the KX500, I was able to pass folks in 4th and 5th gear on the inside line of corners. I must admit that I am a more skilled rider than the others, but as you noted, the 125's turn better, so they should have smoked my ability to get an inside line. Not so. My bike setup, power, and skill were just no match.
Start with bike setup for corners, and then work your way into using the power to help around corners. The '99 KX250 is a great bike, with great power and suspension.
Enjoy.
Rick