The weight of the robot (excepting the basketballs) is supported by the downward-facing ball casters. Basketballs are typically pressurized to 8psi, so assuming the robot weighs 100 lb, you would need to apply that force over 3+ square inches on each ball. As the casters are not that large, the basketball would significantly deform at this point to bear the load, interfering with the drive-ability (though obviating the need for springs). Deformation would get worse under acceleration. I would consider stiffer balls, perhaps like asid61’s suggestion of metal (or rubber over metal) ones, or at least larger load-bearing ball casters.
Edit: This would be even cooler as a kiwi; I think the forces would add up properly for rotation and translation, though less efficiently at the ball-carpet interface.
In either case, It appears that the inverse kinematic equations are to drive each wheel as though it were an omni, but in the opposite direction. Negative Killough/Mecanum for four-ball, negative kiwi for three-ball.