Ball-balancing robots (ballbots) must lean to accelerate, and the lean is limited by the wheel torques and by safety. Planners that use the full robot model are expensive, while planners that ignore the balance dynamics produce references that cannot be followed. This paper presents a two-layer navigation scheme for a three-omni-wheel ballbot. The upper layer is a model predictive controller on a \emph{jerk model} of the ball position, with discrete-time control barrier functions for static, moving, rotating and concave obstacles. The lower layer is a hierarchical sliding mode controller with an extended state observer and a robust exponential control barrier function on the tilt (HSMC-ESO-ECBF). The layers are linked by an acceleration bound derived in closed form from the largest steady tilt that the barrier function admits, so that the plan never asks for a lean the tracker refuses. For obstacles whose motion is only predicted, the obstacles are inflated by conformal prediction radii that bound the prediction error with a prescribed probability. In MuJoCo, with omni-wheel roller contacts and torque saturation, the scheme keeps at least $10$\,cm of clearance, tracks the plan within $2$\,cm RMS, keeps the tilt below the limit and plans in $9$--$12$\,ms, $10$--$15$ times faster than a full-model nonlinear MPC that enters the inflated obstacles; with predicted obstacles, the conformal inflation removes the collisions of a constant-velocity predictor in $30$ random crossing scenarios and reduces them in an endless run on a closed path.The source code and video illustrations are available at https://safe-ballbot.github.io

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