We investigate analytically and numerically the stability of bubble-like fluxons in disk-shaped heterogeneous Josephson junctions. Using ring solitons as a model of bubble fluxons in the two-dimensional sine-Gordon equation, we show that the insertion of coaxial dipole currents prevents their collapse. We characterize the onset of instability by introducing a single parameter that couples the radius of the bubble fluxon with the properties of the injected current. For different combinations of parameters, we report the formation of stable oscillating bubbles, the emergence of internal modes, and bubble breakup due to internal mode instability. We show that the critical germ depends on the ratio between its radius and the steepness of the wall separating the different phases in the system. If the steepness of the wall is increased (decreased), the critical radius decreases (increases). Our theoretical findings are in good agreement with numerical simulations.