We investigate the possible development of magnetohydrodynamical instabilities in the EULAG-MHD millenium simulation of Passos & Charbonneau (2014). This simulation sustains a large-scale magnetic cycle characterized by solar-like polarity reversals taking place on a regular multidecadal cadence, and in which zonally-oriented bands of strong magnetic field accumulate below the convective layers, in response to turbulent pumping from above in successive magnetic half-cycles. Key aspects of this simulation include low numerical dissipation and a strongly subadiabatic fluid layer underlying the convectively unstable layers corresponding to the modeled solar convection zone. These properties are conducive to the growth and development of two-dimensional instabilities otherwise suppressed by stronger dissipation. We find evidence for the action of a non-axisymmetric magnetoshear instability operating in the upper portions of the stably stratified fluid layers. We also investigate the possibility that the Tayler instability may be contributing to the destabilization of the large-scale axisymmetric magnetic component at high latitudes. On the basis of our analyses, we propose a global dynamo scenario whereby the magnetic cycle is driven primarily by turbulent dynamo action in the convecting layers, but MHD instabilities accelerate the dissipation of the magnetic field pumped down into the overshoot and stable layers, thus perhaps significantly influencing the magnetic cycle period. Support for this scenario is found in the distinct global dynamo behaviors observed in an otherwise identical EULAG-MHD simulations, using a different degree of subadiabaticity in the stable fluid layers underlying the convection zone.