Since the core question is actually how to strip the rings of the planet Saturn this answer will ignore a runaway planet Uranus because this has too many problems and may not do the job anyway.
Simply throw a great big dirty snowball at the rings of Saturn and have it moving in the opposite direction of the rotation of Saturn's rings. Big dirty snowballs are better known as comets. This works best if the comet is travelling sufficiently slowly that it falls into orbit around Saturn. It will keep moving in the debris field of the rings. Each successive impact, it send fragments of the rings either spirally down into a lower orbit where there will be further collisions.
Outgassing from the comet will throw up a braking cloud and decelerate ring fragments. Eventually the rings will descend low enough that the outer edges of the atmosphere will further decelerate the rings for their final plunge as meteorites.
Obviously the comet necessary to create this planetary ring clearing Kessler Effect will need to be large and fairly gassy. The cascade will need to be excessive compared to mere space travel blocking Kessler syndrome.