[chapman1]

Since the mass of the ball wouldn't change with age, the difference must come from the "squishiness". More contact between ball and cradle would help reduce the difference in "squish"

After watching your video, I'd say the critical place to add support is at the back of your ball cradle. We experienced this in 2014. As the catapult accelerates, centrifugal force wants to move the ball away from the fulcrum and up over the back lip of the cradle. Balls of differing "squishiness" will behave differently going over the lip, and if the last point of contact is the lip rather than the full cradle, ball differences will be magnified. If you can add support to limit that roll away from the fulcrum, I bet your launch differences will go away.

If your cradle cannot be made larger, try making it more solid and tipping it toward the fulcrum to reduce "roll away".