In a sense, this "ping pong" is what cycler orbits attempt to do, you are just overestimating the power of flybys.
To make the distance between Jupiter and Saturn in just 4 years (one way?), you're doing the trip often enough that you also have to a bounce when the planets are on opposite sides of the Sun, almost 15 AU. Even in a straight line, that requires travelling at about 70km/s.
What limits flybys is the turning angle $\theta$.
If one slowly enters a system, one can bend the trajectory nearly 180º around. But as one enters the system with some velocity $v_{\infty}$, this angle shrinks, until one just flies past the planet so fast that the trajectory is barely bent.
This angle also depends on the mass parameter of the planet $\mu$ and the flyby periapsis distance $r_P$ at the very least limited by the radius of the planet
$$\theta =2\sin ^{ -1 }{ \left( \frac { 1 }{ 1+\frac { r_P{ v }_{ \infty }^{ 2 } }{ \mu } } \right) } $$
Entering Jupiter and Saturn at 70km/s means bending the trajectories no more than 31º and 13.5º respectively, far too little to turn the spacecraft around to nearly the same direction as it came from. Especially the Saturn end is limited, requiring entry velocities no higher than 16km/s to do a 90º degree turn, which even then is still too small an angle for anything near a rapid bounce between the planets.
Cycler orbits at best do a cycle for every synodic period of the two planets they serve
$$T_{synodic} = \frac{1}{\frac{1}{T_1} - \frac{1}{T_2}}$$
Or some integer multiple of this. For Jupiter-Saturn that's close to 20 years.
