The gravoviscothermal instability dynamics excitable in a spherical composite astrofluidic system (CAS) existent in dwarf satellite galaxies, which are mainly constituted of an admixture of neutral gas fluid (NGF) and cosmic ray fluid (CRF) coupled via a barotropic correlation in spherical geometry, is studied. A spherical normal mode analysis over the perturbed CAS yields a unique quartic dispersion relation having an atypical set of multiparametric coefficients. The growth behaviours of the instability are illustratively specified in two distinct situations: inviscid (ideal) and viscid (real). It is seen that the CAS kinematic viscosity and temperature play stabilizing roles to the CAS instability against the gravity. In contrast, the CRF diffusion destabilizes the instability. A faster rate of structure formation, which is sourced by the CRF moderation effects, is interestingly a formal picture in such astrosituations. It achieves a peaky growth in the acoustic domain at near the centre of the entire CAS mass distribution. The average instability pattern behaviours are fairly confirmed in two distinct pure fluctuation regimes: gravitational (low-frequency, Jeans-type) and the thermal (high-frequency, acoustic-type). The reliability of our results is marginally validated in the light of the prior reports. We lastly offer a tentative application of our main results in the real astrocosmic context of reorganizing and restructuring the astrocloud stability dynamics via the excitation of gravothermal supercritical catastrophic instability, pair-instability, etc.