Dynamic redistribution of internal mass for a more-than-rigid shell
The "star" has to be heavy so as not to move around. But the weight is not distributed uniformly in the interior, nor in the shell, like we assume for many everyday objects.
Instead, there is a small, very dense mass (blue in the illustration) inside the larger "star". This mass is tethered in many directions to the outer faces. When a contestant lands on the star, pushing it, the tethering strands are stretched. Ordinarily, this would make the "star" act as a rigid body, moving slightly in response to the impact. However, in this case, the tether detects the tension from the outside contact, and compensates by increasing that tension with its own motor in near real time (yellow arrows on the gray lines) All linear and any angular momentum are transferred into the central mass.
The result is that the outer shell of the "star" moves scarcely at all, even though it absorbs the contestant's momentum. The momentum is now hidden in the internal motion of the dense inner mass.
The "star" is limited in the amount of momentum it can absorb this way, multiplied by the time. But of course it is also limited by the force of impact it absorbs before it crumples to avoid lethally injuring the contestant. I hope. Given the rules and gameplay, it may be designed so that the internal masses can be reset between matches, or it may have an unobtrusive propulsion system (such as a cable it can launch at the nearest "star" in the right direction, or to a nearby wall) that abruptly transfers the momentum when the action of the game has moved elsewhere. In extremis, the "star" can abruptly move its outer shell so that the mass inside is now heading toward the center; this would have to be included in the rules, since it affects game play.
Minutiae: because the negative feedback on the "star's" motion is accomplished by a positive feedback on the cable linking shell and core, you might think that this would be unstable. However, the system knows how much tension it has added to the system, and can subtract that to understand the true force on the outer shell. Also, the system only needs to add a small percentage to the tension normally needed to accelerate the inner core in order to ensure all the added momentum goes to that core. (The outer hair cells of the ear do this to a much more remarkable extent, greatly amplifying the sounds we hear) Also, because the core needs to absorb angular momentum, it may contain reaction wheels that adjust the core so that it doesn't literally need to rotate while remaining attached to its network of cables.