Based on my simulation of Maxwell's equations in cylindrical coordinates, a stable oscillation of the magnetic field does not exist. I allowed for general evolution of the field (i.e., aside from avoiding non-physical singularities, there were no constraints on the behavior), and ultimately the fields formed a wave which escaped the system at the speed of light. This is essentially what I originally suspected, but it was an interesting problem to solve.
It might be argued that I did not form the correct initial conditions. Indeed some small artifacts are seen in the movie which represent very small imperfections in the initial model. However these artifacts should not be an issue. Maxwell's equations are linear, so the addition of a small correction term to remove the artifacts will lead to only a small correction to the final solution. Also, I argue that if the formation of a putatively stable flux loop is so sensitive to the initial conditions, then one could probably never form in nature. (i.e., the waves would form and propagate outward before the ``perfect'' stable configuration -- if it exists -- could arise).
Immediately after starting the computations, the magnetic field begins to change from it's initial flux-loop configuration. From the above expression, a poloidal electric field is generated, however, the oscillations never return to the initial configuration. By the time that the electric field component has become dominant at the original position of the flux loop, the magnetic field component has propagated one half-cycle outward. From there it is clear that the propagation will continue the system will never return to its initial configuration.